Recent zbMATH articles in MSC 05https://zbmath.org/atom/cc/052023-09-22T14:21:46.120933ZWerkzeugAlgebras of binary isolating formulas for theories of root products of graphshttps://zbmath.org/1517.030352023-09-22T14:21:46.120933Z"Emel'yanov, Dmitriĭ Yur'evich"https://zbmath.org/authors/?q=ai:emelyanov.dmitrii-yurevichSummary: Algebras of distributions of binary isolating and semi-isolating formulas are derived objects for given theory and reflect binary formula relations between realizations of 1-types. These algebras are associated with the following natural classification questions: 1) for a given class of theories, determine which algebras correspond to the theories from this class and classify these algebras; 2) to classify theories from a given class depending on the algebras defined by these theories of isolating and semi-isolating formulas. Here the description of a finite algebra of binary isolating formulas unambiguously entails a description of the algebra of binary semi-isolating formulas, which makes it possible to track the behavior of all binary formula relations of a given theory. The paper describes algebras of binary formulae for root products. The Cayley tables are given for the obtained algebras. Based on these tables, theorems describing all algebras of binary formulae distributions for the root multiplication theory of regular polygons on an edge are formulated. It is shown that they are completely described by two algebras.Graph theory in America. The first hundred yearshttps://zbmath.org/1517.050012023-09-22T14:21:46.120933Z"Wilson, Robin"https://zbmath.org/authors/?q=ai:wilson.robin-j"Watkins, John J."https://zbmath.org/authors/?q=ai:watkins.john-j"Parks, David J."https://zbmath.org/authors/?q=ai:parks.david-jLet us begin with the following description:
``\textit{Graph Theory in America} focuses on the development of graph theory in North America from 1876 to 1976. At the beginning of this period, James Joseph Sylvester, perhaps the finest mathematician in the English-speaking world, took up his appointment as the first professor of mathematics at the Johns Hopkins University, where his inaugural lecture outlined connections between graph theory, algebra, and chemistry -- shortly after, he introduced the word graph in our modern sense. A hundred years later, in 1976, graph theory witnessed the solution of the long-standing four color problem by Kenneth Appel and Wolfgang Haken of the University of Illinois.
Tracing graph theory's trajectory across its first century, this book looks at influential figures in the field, both familiar and less known. Whereas many of the featured mathematicians spent their entire careers working on problems in graph theory, a few such as Hassler Whitney started there and then moved to work in other areas. Others, such as C. S. Peirce, Oswald Veblen, and George Birkhoff, made excursions into graph theory while continuing their focus elsewhere. Between the main chapters, the book provides short contextual interludes, describing how the American university system developed and how graph theory was progressing in Europe. Brief summaries of specific publications that influenced the subject's development are also included.
\textit{Graph Theory in America} tells how a remarkable area of mathematics landed on American soil, took root, and flourished.''
In addition to the publisher's description, one can note the following.
In the first section, entitled as ``Setting the scene: early American mathematics'', the focus is on the first institutions of higher education which were established in the American colonies. Some additional descriptions about Harvard University, Yale University, and Princeton University, as well as the Massachusetts Institute of Technology (MIT) and Johns Hopkins University are given. Also, some peculiarities of the development of mathematics education in the early years in the USA are considered, as well as the contributions of Benjamin Pierce and Eliakim Hastings Moore to this development are briefly described.
Chapter 1 ``The 1800s'' is devoted to the history of mathematical research under the influence of James Joseph Sylvester and Johns Hopkins University, as well as to the early interest in graph theory in America and to some scientists. The American Journal of Mathematics is mentioned as the oldest mathematics journal in continuous publication in North America. Special attention is also given to the four-color theorem.
The titles of the following chapters are: ``The 1900s and 1910s'', ``The 1920s'', ``The 1930s'', ``The 1940s and 1950s'', ``The 1960s and 1970s''. In these chapters, the attention mainly is given to peculiarities of the development of investigations, to some mathematicians (brief biographical data, descriptions of results, publications, and some proofs, etc.), as well as to certain conjectures, algorithms, and to explanations of several notions. Several problems, including a progress in solving the four-color problem, are discussed. The development of mathematics science is also briefly considered in the periods of the Great Depression, World War I, and World War II. The Zentralblatt für Mathematik and the Mathematical Reviews are reported on.
Reviewer: Symon Serbenyuk (Kyjiw)Catalan and Schröder permutations sortable by two restricted stackshttps://zbmath.org/1517.050022023-09-22T14:21:46.120933Z"Baril, Jean-Luc"https://zbmath.org/authors/?q=ai:baril.jean-luc"Cerbai, Giulio"https://zbmath.org/authors/?q=ai:cerbai.giulio"Khalil, Carine"https://zbmath.org/authors/?q=ai:khalil.carine"Vajnovszki, Vincent"https://zbmath.org/authors/?q=ai:vajnovszki.vincentSummary: Pattern avoiding machines were introduced recently by \textit{G. Cerbai} et al. [J. Comb. Theory, Ser. A 173, Article ID 105230, 19 p. (2020; Zbl 1435.05004)] as a particular case of the two-stacks in series sorting device. They consist of two restricted stacks in series, ruled by a right-greedy procedure and the stacks avoid some specified patterns. Some of the obtained results have been further generalized to Cayley permutations by Cerbai [loc. cit.], specialized to particular patterns by \textit{C. Defant} and \textit{K. Zheng} [Adv. Appl. Math. 128, Article ID 102192, 33 p. (2021; Zbl 1467.05003)], or considered in the context of functions over the symmetric group by \textit{K. Berlow} [Discrete Math. 344, No. 11, Article ID 112571, 11 p. (2021; Zbl 1472.05004)]. In this work we study pattern avoiding machines where the first stack avoids a pair of patterns of length 3 and investigate those pairs for which sortable permutations are counted by the (binomial transform of the) Catalan numbers and the Schröder numbers.Descent distribution on Catalan words avoiding ordered pairs of relationshttps://zbmath.org/1517.050032023-09-22T14:21:46.120933Z"Baril, Jean-Luc"https://zbmath.org/authors/?q=ai:baril.jean-luc"Ramírez, José L."https://zbmath.org/authors/?q=ai:ramirez.jose-luisSummary: This work is a continuation of some recent articles presenting enumerative results for Catalan words avoiding one or a pair of consecutive or classical patterns of length 3. More precisely, we provide systematically the bivariate generating function for the number of Catalan words avoiding a given pair of relations with respect to the length and the number of descents. We also present several constructive bijections preserving the number of descents. As a byproduct, we deduce the generating function for the total number of descents on all Catalan words of a given length and avoiding a pair of ordered relations.Maxima and visibility in involutionshttps://zbmath.org/1517.050042023-09-22T14:21:46.120933Z"Barnabei, Marilena"https://zbmath.org/authors/?q=ai:barnabei.marilena"Bonetti, Flavio"https://zbmath.org/authors/?q=ai:bonetti.flavio"Castronuovo, Niccolò"https://zbmath.org/authors/?q=ai:castronuovo.niccolo"Silimbani, Matteo"https://zbmath.org/authors/?q=ai:silimbani.matteoSummary: We enumerate involutions according to the joint distribution of left-to-right and right-to-left maxima. From this computation we deduce the distribution of the static ``number of visible pairs'', namely, the number of non-adjacent columns in the bargraph representation of a given permutation that are mutually visible to each other. The corresponding generating functions are expressible in terms of formal continued fractions. We obtain the same distribution over the set of involutions avoiding either the pattern 3412 or 4321. The proofs reside on a well-known bijection between involutions and labeled Motzkin paths.A short note on polynomials \(f(X)=X+A X^{1 + q^2 (q - 1) / 4}+B X^{1 + 3 q^2 (q - 1) / 4} \in \mathbb{F}_{q^2}[X]\), \(q\) evenhttps://zbmath.org/1517.050052023-09-22T14:21:46.120933Z"Bartoli, Daniele"https://zbmath.org/authors/?q=ai:bartoli.daniele"Bonini, Matteo"https://zbmath.org/authors/?q=ai:bonini.matteoSummary: An alternative proof of the necessary conditions on \(A,B \in \mathbb{F}_{q^2}^\ast\) for \(f(X)=X+A X^{1 + q^2 (q - 1) / 4}+B X^{1 + 3 q^2 (q - 1) / 4}\) to be a permutation polynomial in \(\mathbb{F}_{q^2}\), \(q\) even, is given. This proof involves standard arguments from algebraic geometry over finite fields and fast symbolic computations.Permutations avoiding 4321 and 3241 have an algebraic generating functionhttps://zbmath.org/1517.050062023-09-22T14:21:46.120933Z"Callan, David"https://zbmath.org/authors/?q=ai:callan.davidSummary: We show that permutations avoiding both of the (classical) patterns 4321 and 3241 have the algebraic generating function conjectured by Vladimir Kruchinin.The number of \(\{1243, 2134\}\)-avoiding permutationshttps://zbmath.org/1517.050072023-09-22T14:21:46.120933Z"Callan, David"https://zbmath.org/authors/?q=ai:callan.davidSummary: We show that the counting sequence for permutations avoiding both of the (classical) patterns 1243 and 2134 has the algebraic generating function supplied by Vaclav Kotesovec for sequence A164651 in The On-Line Encyclopedia of Integer Sequences..The \(h^\ast\)-polynomial of the order polytope of the zig-zag posethttps://zbmath.org/1517.050082023-09-22T14:21:46.120933Z"Coons, Jane Ivy"https://zbmath.org/authors/?q=ai:coons.jane-ivy"Sullivant, Seth"https://zbmath.org/authors/?q=ai:sullivant.sethSummary: We construct a family of shellings for the canonical triangulation of the order polytope of the zig-zag poset. This gives a new combinatorial interpretation for the coefficients in the numerator of the Ehrhart series of this order polytope in terms of the swap statistic on alternating permutations. We also offer an alternate proof of this result using the techniques of rank selection. Finally, we show that the sequence of coefficients of the numerator of this Ehrhart series is symmetric and unimodal.Classification of maps on a finite set under permutationhttps://zbmath.org/1517.050092023-09-22T14:21:46.120933Z"García Zapata, Juan Luis"https://zbmath.org/authors/?q=ai:garcia-zapata.juan-luis"Rico-Gallego, Juan-Antonio"https://zbmath.org/authors/?q=ai:rico-gallego.juan-antonioSummary: We characterize when two maps \(f,g: X \rightarrow X\) on a finite set are conjugated, that is, when there is a permutation \(\sigma: X \rightarrow X\) such that \(f= \sigma^{-1} g \sigma\). We build a signature \(\operatorname{sgn}(f)\) for every map \(f\) such that \(f\) and \(g\) are conjugated if and only if \(\operatorname{sgn}(f)= \operatorname{sgn}(g)\). This signature is an array that includes information about the cycles of the map and the noncyclic elements, called transients, besides data about the insertion of the transients in the cycle. The transient elements form several tree shape graphs. Our characterization is a generalization of the known fact that two permutations are conjugated if and only if they have the same cycle structure. We use elementary facts about finite dynamical systems and about the canonical labeling of unordered trees. The signature can be built by an algorithm of complexity \(O(n)\), being \(n\) the cardinality of \(X\).Decomposing random permutations into order-isomorphic subpermutationshttps://zbmath.org/1517.050102023-09-22T14:21:46.120933Z"Groenland, Carla"https://zbmath.org/authors/?q=ai:groenland.carla"Johnston, Tom"https://zbmath.org/authors/?q=ai:johnston.tom"Korandi, Daniel"https://zbmath.org/authors/?q=ai:korandi.daniel"Roberts, Alexander"https://zbmath.org/authors/?q=ai:roberts.alexander"Scott, Alex"https://zbmath.org/authors/?q=ai:scott.alexander-d"Tan, Jane"https://zbmath.org/authors/?q=ai:tan.janeSummary: Two permutations \(\sigma\) and \(\pi\) are \(\ell\)-similar if they can be decomposed into subpermutations \(\sigma^{(1)},\dots,\sigma^{(\ell)}\) and \(\pi^{(1)},\dots,\pi^{(\ell)}\) such that \(\sigma^{(i)}\) is order-isomorphic to \(\pi^{(i)}\) for all \(i\in[\ell]\). Recently, \textit{A. Dudek} et al. [Electron. J. Comb. 28, No. 3, Research Paper P3.19, 18 p. (2021; Zbl 1470.05009)] posed the problem of determining the minimum \(\ell\) for which two permutations chosen independently and uniformly at random are \(\ell\)-similar. We show that two such permutations are \(O(n^{1/3}\log^{11/6}(n))\)-similar with high probability, which is tight up to a polylogarithmic factor. Our result also generalizes to simultaneous decompositions of multiple permutations.\((M,i)\)-multiset Eulerian polynomialshttps://zbmath.org/1517.050112023-09-22T14:21:46.120933Z"Ma, Jun"https://zbmath.org/authors/?q=ai:ma.jun.2|ma.jun"Pan, Kaiying"https://zbmath.org/authors/?q=ai:pan.kaiyingSummary: Let \(M = \{ 1^{p_1}, \ldots, n^{p_n} \}\) be a multiset, \( \mathfrak{S}_{M, i}\) be the set of multipermutations over \(M\) with \(i\) as the first entry and \(A_{M, i}(x)\) be the enumerators of descents over \(\mathfrak{S}_{M,i}\). \(A_{M,i}(x)\) is called the \((M, i)\)-multiset Eulerian polynomial. A Carlitz-type identity of \(A_{M,i}(x)\) is derived. It is proved that \(x A_{M,i}(x)\), \(c_1 A_{M,i}(x)+c_2 A_{M,j}(x)\) and \(c_1 x A_{M,i}(x) + c_2 A_{M,j}(x)\) have only real roots, where \(c_1\) and \(c_2\) are nonnegative real number, \(i, j \in M\) and \(i < j\). For the multiset \(M = \{ 1^k, 2^k, \ldots, n^k \} \), it is shown that \(A_{M,i}(x)\) is reciprocal with \(A_{M, n - i + 1}(x)\), \(A_{M,i}(x) + A_{M,n-i+1}(x)\) and \(x A_{M,i}(x) + A_{M,n - i + 1}(x)\) are \(\gamma \)-positive, and \(A_{M, i}(x)\) is bi-\( \gamma \)-positive for any \(1 \leq i \leq \frac{ n + 1}{ 2} \). For \(M = \{1, 2, \ldots, n \}\) and \(1 \leq i \leq n\), we give a combinatorial interpretation for \(\gamma \)-coefficients of \(A_{M,i}(x) + A_{M,n - i + 1}(x)\).Cycles of even-odd drop permutations and continued fractions of Genocchi numbershttps://zbmath.org/1517.050122023-09-22T14:21:46.120933Z"Pan, Qiongqiong"https://zbmath.org/authors/?q=ai:pan.qiongqiong"Zeng, Jiang"https://zbmath.org/authors/?q=ai:zeng.jiangSummary: Recently \textit{A. Lazar} and \textit{M. L. Wachs} [Comb. Theory 2, No. 1, Paper No. 2, 34 p. (2022; Zbl 1502.52019)] proved two new permutation models, called D-permutations and E-permutations, for Genocchi and median Genocchi numbers. In a follow-up, \textit{S.-P. Eu} et al. [Electron. J. Comb. 29, No. 2, Research Paper P2.15, 23 p. (2022; Zbl 1487.05011)] studied the even-odd descent permutations, which are in bijection with E-permutations. We generalize Eu et al.'s descent polynomials with eight statistics and obtain an explicit J-fraction formula for their ordinary generaing function. The J-fraction permits us to confirm two conjectures of Lazar-Wachs [loc. cit.] about cycles of D and E permutations and obtain a \((p,q)\)-analogue of Eu et al.'s gamma-formula [loc. cit.]. Moreover, the \((p,q)\) gamma-coefficients have the same factorization flavor as the gamma-coefficients of \textit{P. Brändén}'s \((p,q)\)-Eulerian polynomials [in: Handbook of enumerative combinatorics. Boca Raton, FL: CRC Press. 437--483 (2015; Zbl 1327.05051)].A unified approach to generalized Pascal-like matrices: \(q\)-analysishttps://zbmath.org/1517.050132023-09-22T14:21:46.120933Z"Akkus, Ilker"https://zbmath.org/authors/?q=ai:akkus.ilker"Kizilaslan, Gonca"https://zbmath.org/authors/?q=ai:kizilaslan.gonca"Verde-Star, Luis"https://zbmath.org/authors/?q=ai:verde-star.luisSummary: In this paper, we present a general method to construct \(q\)-analogues and other generalizations of Pascal-like matrices. Our matrices are obtained as functions of strictly lower triangular matrices and include several types of generalized Pascal-like matrices and matrices related with modified Hermite polynomials of two variables and other polynomial sequences. We find explicit expressions for products, powers, and inverses of the matrices and also some factorization formulas using this method.Enumerating two permutation classes by the number of cycleshttps://zbmath.org/1517.050142023-09-22T14:21:46.120933Z"Archer, Kassie"https://zbmath.org/authors/?q=ai:archer.kassieSummary: We enumerate permutations in the two permutation classes \(\mathrm{Av}_n(312, 4321)\) and \(\mathrm{Av}_n(321, 4123)\) by the number of cycles each permutation admits. We also refine this enumeration with respect to several statistics.Matching points in compositions and wordshttps://zbmath.org/1517.050152023-09-22T14:21:46.120933Z"Archibald, Margaret"https://zbmath.org/authors/?q=ai:archibald.margaret"Blecher, Aubrey"https://zbmath.org/authors/?q=ai:blecher.aubrey"Knopfmacher, Arnold"https://zbmath.org/authors/?q=ai:knopfmacher.arnoldSummary: A matching point in compositions and words is an extension to these objects of the well-studied concept of fixed points in permutations. The equivalent of the derangement problem is solved here by providing a formula for the number of compositions of \(n\) having no matching points, and showing that the number of words with no matching points tends to zero as (the length of the word) \(n\) tends to infinity. We also find formulae for the average number of matching points in both words and compositions.Baxter permuton and Liouville quantum gravityhttps://zbmath.org/1517.050162023-09-22T14:21:46.120933Z"Borga, Jacopo"https://zbmath.org/authors/?q=ai:borga.jacopo"Holden, Nina"https://zbmath.org/authors/?q=ai:holden.nina"Sun, Xin"https://zbmath.org/authors/?q=ai:sun.xin"Yu, Pu"https://zbmath.org/authors/?q=ai:yu.puSummary: The Baxter permuton is a random probability measure on the unit square which describes the scaling limit of uniform Baxter permutations. We determine an explicit formula for the density of the expectation of the Baxter permuton. This answers a question of \textit{T. Dokos} and \textit{I. Pak} [Online J. Anal. Comb. 9, Article 5, 12 p. (2014; Zbl 1292.05022)]. We also prove that all pattern densities of the Baxter permuton are strictly positive, distinguishing it from other permutons arising as scaling limits of pattern-avoiding permutations. Our proofs rely on a recent connection between the Baxter permuton and Liouville quantum gravity (LQG) coupled with the Schramm-Loewner evolution (SLE). The method works equally well for a two-parameter generalization of the Baxter permuton recently introduced by the first author, except that the density is not as explicit. This new family of permutons, called skew Brownian permuton, describes the scaling limit of a number of random constrained permutations. We finally observe that in the LQG/SLE framework, the expected proportion of inversions in a skew Brownian permuton equals \(\frac{\pi -2\theta }{2\pi }\) where \(\theta\) is the so-called imaginary geometry angle between a certain pair of SLE curves.Linked partition ideals and Euclidean billiard partitionshttps://zbmath.org/1517.050172023-09-22T14:21:46.120933Z"Chern, Shane"https://zbmath.org/authors/?q=ai:chern.shaneThis paper considers recently introduced Euclidean billiard partitions, that arose in the study of integrable billiards in Euclidean spaces of arbitrary dimensions and represent the winding numbers for each elliptic coordinate corresponding to periodic trajectories. Such partitions have distinct parts such that the adjacent ones are never both odd and the smallest part is even. This work uses linked partition ideals to establish several relevant trivariate generating function identities, which then enable a new way of obtaining the generating functions for the billiard partitions.
Reviewer: Milena Radnović (Sydney)The number of disordered covers of a finite set by subsets having fixed cardinalitieshttps://zbmath.org/1517.050182023-09-22T14:21:46.120933Z"Ganopol'skiĭ, R. M."https://zbmath.org/authors/?q=ai:ganopolskii.r-mSummary: This article describes a new type of combinatorial numbers which calculate amount of the covers of a finite set by subsets having fixed cardinalities -- parameters of numbers. A series of relations and identities are proved for them. Some sums of these numbers are computed. Special cases of new combinatorial numbers with parameters satisfying certain relations are investigated. Several other applications of these numbers in discrete mathematics are shown.A combinatorial problem and its application to probability theory. Ihttps://zbmath.org/1517.050192023-09-22T14:21:46.120933Z"Narayana, T. V."https://zbmath.org/authors/?q=ai:narayana.tadepalli-venkataSummary: We solve a combinatorial problem which generalizes the `problème du scrutin' of D. André. In a particular case, this result may be interpreted as a quasi-order defined on the \(r\)-partitions of an integer. We indicate the relation of these quasi-orderings to certain coin tossing problems in probability theory considered by the author in a previous paper [C. R. Acad. Sci., Paris 240, 1188--1189 (1955; Zbl 0064.12705)].A note on difference equations and combinatorial identities arising out of coin tossing problemshttps://zbmath.org/1517.050202023-09-22T14:21:46.120933Z"Narayana, T. V."https://zbmath.org/authors/?q=ai:narayana.tadepalli-venkata"Mohanty, S. G."https://zbmath.org/authors/?q=ai:mohanty.sri-gopalSummary: We solve a class of difference equations and desire some combinatorial identities arising from ``returns to equilibrium'' in coin tossing problems. We shall use the results and the notations introduced by the senior author in three previous papers which are referred to in what follows as (1), (2) and (3).A combinatorial problem and its applications to probability theory. IIhttps://zbmath.org/1517.050212023-09-22T14:21:46.120933Z"Narayana, T. V."https://zbmath.org/authors/?q=ai:narayana.tadepalli-venkata"Mohanty, S. G."https://zbmath.org/authors/?q=ai:mohanty.sri-gopal"Ladouceur, J. C."https://zbmath.org/authors/?q=ai:ladouceur.j-cSummary: Games \(A_{2k}\), \(A_{2k+1}\) arising out of tossing two coins with probabilities \(p_1\), \(p_2\) of obtaining heads where \(p_1 + p_2 > 1\) are defined in the paper. Some of the properties of these games are derived and utilised to obtain and solve a few difference equations. The solution of the difference equations, on the other hand, leads to some interesting identities in probability theory, Besides these results, the generating function of \(A_{2k}\) is obtained from which the duration of the game and its variance are also derived.
For Part I see [the first author, ibid. 7, No. 1--2, 169--178 (1955; Zbl 1517.05019)].On a generalized basic series and Rogers-Ramanujan type identities. IIhttps://zbmath.org/1517.050222023-09-22T14:21:46.120933Z"Sonik, P."https://zbmath.org/authors/?q=ai:sonik.p"Goyal, Megha"https://zbmath.org/authors/?q=ai:goyal.meghaSummary: This paper is in continuation with our recent paper ``On a generalized basic series and Rogers-Ramanujan type identities'' [the first author et al., Contrib. Discrete Math. 18, No. 1, 15--28 (2023; Zbl 1517.05023)]. Here, we consider two generalized basic series and interpret these basic series as the generating function of some restricted \((n + t)\)-color partitions and restricted weighted lattice paths. The basic series discussed in the aforementioned paper, is now a mere particular case of one of the generalized basic series that are discussed in this paper. Besides, eight particular cases are also discussed which give combinatorial interpretations of eight Rogers-Ramanujan type identities which are combinatorially unexplored till date.On a generalized basic series and Rogers-Ramanujan type identitieshttps://zbmath.org/1517.050232023-09-22T14:21:46.120933Z"Sonik, P."https://zbmath.org/authors/?q=ai:sonik.p"Ranganatha, D."https://zbmath.org/authors/?q=ai:ranganatha.d"Goyal, Megha"https://zbmath.org/authors/?q=ai:goyal.meghaSummary: In this paper, we give the generalization of MacMahon's type combinatorial identities. A generalized \(q\)-series is interpreted as the generating function of two different combinatorial objects, viz., restricted \(n\)-color partitions and weighted lattice paths which give entirely new Rogers-Ramanujan-MacMahon type combinatorial identities. This result yields an infinite class of 2-way combinatorial identities which further extends the work of Agarwal and Goyal. We also discuss the bijective proof of the main result. Forbye, eight particular cases are also discussed which give a combinatorial interpretation of eight entirely new Rogers-Ramanujan type identities.The multiplicity of left-to-right maxima in geometrically distributed wordshttps://zbmath.org/1517.050242023-09-22T14:21:46.120933Z"Archibald, M."https://zbmath.org/authors/?q=ai:archibald.margaret"Knopfmacher, A."https://zbmath.org/authors/?q=ai:knopfmacher.arnoldThe paper studies weak left-to-right maxima in geometric words. A geometric word of length \(n\) is a sequence of \(n\) i.i.d geometric random variables where the probability that the \(j\)-th term equals \(i\) is \((1-p)^{i-1}p\). The authors determine the number of values that occur exactly \(m\) times as a weak left-to-right maximum. For \(a_1, \dots, a_n\) one says that \(a_j\) is a weak left-to-right maximum \(a_j \ge a_i\) for each \(1 \le i < j \). The actual results are too complex to state here.
Reviewer: Wolfgang A. Schmid (Paris)Beck-type companion identities for Franklin's identityhttps://zbmath.org/1517.050252023-09-22T14:21:46.120933Z"Ballantine, Cristina"https://zbmath.org/authors/?q=ai:ballantine.cristina-m"Welch, Amanda"https://zbmath.org/authors/?q=ai:welch.amandaSummary: In 2017, Beck conjectured that the difference in the number of parts in all partitions of \(n\) into odd parts and the number of parts in all strict partitions of \(n\) is equal to the number of partitions of \(n\) whose set of even parts has one element, and also to the number of partitions of \(n\) with exactly one part repeated. Andrews proved the conjecture using generating functions. Beck's identity is a companion identity to Euler's identity. The theorem has been generalized (with a combinatorial proof) by Yang to a companion identity to Glaisher's identity. Franklin generalized Glaisher's identity, and in this article, we provide a Beck-type companion identity to Franklin's identity and prove it both analytically and combinatorially. Andrews' and Yang's respective theorems fit naturally into this very general frame. We also discuss how Franklin's identity and the companion Beck-type identities can be further generalized to Euler pairs of any order.Several properties of differential equation with \((p, q)\)-Genocchi polynomialshttps://zbmath.org/1517.050262023-09-22T14:21:46.120933Z"Kang, J. Y."https://zbmath.org/authors/?q=ai:kang.jiayi|kang.jianying|kang.jung-yoog|kang.ji-yoon|kang.jiayu|kang.jingyu|kang.jung-yup|kang.jiayin|kang.jiyangSummary: We construct several differential equations of which are related to \((p, q)\)-Genocchi polynomials in this paper. From these differential equation, we also investigate some relations which are related to Genocchi, \(q\)-Genocchi, and \((p, q)\)-Genocchi polynomials.Representing the Stirling polynomials \(\sigma_n(x)\) in dependence of \(n\) and an application to polynomial zero identitieshttps://zbmath.org/1517.050272023-09-22T14:21:46.120933Z"Kovačec, Alexander"https://zbmath.org/authors/?q=ai:kovacec.alexander"de Tovar Sá, Pedro Barata"https://zbmath.org/authors/?q=ai:de-tovar-sa.pedro-barataSummary: Denote by \(\sigma_n\) the \(n\)-th Stirling polynomial in the sense of the well-known book [Concrete mathematics. A foundation for computer science. Reading, MA: Addison-Wesley Publishing Company (1989; Zbl 0668.00003)] by \textit{R. L. Graham} et al.. We show that there exist developments \(x{\sigma}_n(x)={\sum}_{j=0}^n{({2}^j j!)}^{-1}{q}_{n-j}(j){x}^j\) with polynomials \(q_j\) of degree j. We deduce from this the polynomial identities
\[
\sum_{a+b+c+d=n}{(-1)}^d\frac{{(x-2a-2b)}^{3n-s-a-c}}{a!b!c!d!(3n-s-a-c)!}=0, \quad \text{for } s\in{{\mathbb{Z}}}_{\ge 1},
\]
found in an attempt to find a simpler formula for the density function in a five-dimensional random flight problem. We point out a probable connection to Riordan arrays.Bivariate extension of the \(r\)-Dowling polynomials and two forms of generalized Spivey's formulahttps://zbmath.org/1517.050282023-09-22T14:21:46.120933Z"Mangontarum, Mahid M."https://zbmath.org/authors/?q=ai:mangontarum.mahid-mSummary: We extend the notion of \(r\)-Dowling polynomials to their bivariate forms and establish several properties that generalize those of the bivariate Bell and \(r\)-Bell polynomials. Lastly, we obtain two forms of generalized Spivey's formula.Properties of \((p, q)\)-differential equations with \((p, q)\)-Euler polynomials as solutionshttps://zbmath.org/1517.050292023-09-22T14:21:46.120933Z"Yu, C. H."https://zbmath.org/authors/?q=ai:yu.chunhui|yu.cheng-he|yu.chunhai|yu.chenghai|yu.changhua|yu.ching-hao|yu.ching-hua|yu.cheng-hao|yu.caihua|yu.chiew-hui|yu.chien-hsien|yu.cihai|yu.chaohang|yu.chao-hua|yu.chonghua|yu.chang-hsi|yu.chenhai|yu.canghai|yu.chung-hyo|yu.chung-hsin|yu.chi-hua|yu.chung-hyun|yu.cunhai|yu.chong-ho|yu.changhui|yu.cheng-han|yu.chunhua|yu.caihui|yu.chenghui|yu.chuanhua"Kang, J. Y."https://zbmath.org/authors/?q=ai:kang.jiayin|kang.jiayu|kang.ji-yoon|kang.jung-yup|kang.jiyang|kang.jingyu|kang.jianying|kang.jung-yoog|kang.jiayiSummary: In this paper, we discuss \((p, q)\)-differential equations which are related to \((p, q)\)-Euler polynomials. Also, we find a basic symmetric property for \((p, q)\)-differential equation using the generating function of \((p, q)\)-Euler polynomials.A simple proof of higher order Turán inequalities for Boros-Moll sequenceshttps://zbmath.org/1517.050302023-09-22T14:21:46.120933Z"Zhao, James Jing Yu"https://zbmath.org/authors/?q=ai:zhao.james-jing-yuThis paper is a different approach to a recent result concerning higher-order Turán inequalities for the Boros-Moll sequence \(\{d_l(m)\}_{l=0}^m\) obtained by \textit{J. J. F. Guo} [J. Number Theory 225, 294--309 (2021; Zbl 1465.05017)]. Here, \(d_l\) is the coefficient of \(x^l\) in the Boros-Moll polynomials
\[
P_m(x)=\sum_{j,k}\binom{2m+1}{2j}\binom{m-j}{k}\binom{2k+2j}{k+j}\frac{(x+1)^j(x-1)^k}{2^{3(k+j)}}.
\]
These polynomials arise in the study of a quartic integral
\[
\int_0^\infty \frac{dt}{{(t^4+2xt^2+1)}^{m+1}}=\frac{\pi}{2^{m+3/2}(x+1)^{m+1/2}}P_m(x).
\]
The paper discusses, among other things, a sharper bound for \(\frac{d_l(m+1)}{d_l(m)}\), and one for \(\frac{d_l(m)^2}{d_{l-1}(m)d_{l+1}(m)}\).
Reviewer: Firdous Ahmad Mala (Srinagar)Constant and nearly constant block-sum partially balanced incomplete block designs and magic rectangleshttps://zbmath.org/1517.050312023-09-22T14:21:46.120933Z"Khattree, Ravindra"https://zbmath.org/authors/?q=ai:khattree.ravindraIn this paper, the author establishes a connection between magic rectangles and constant block-sum partially balanced incomplete block designs.
A magic rectangle of order \(a_1\times a_2\) is an array in elements \(1, 2,\dots, a_1a_2\), each appearing only once so that each of the \(a_1\) rows adds to a constant \(A_1\) and each of the \(a_2\) columns adds to a constant \(A_2\). A magic rectangle of order \(a_1\times a_2\) exists if and only if \(a_1\) and \(a_2\) have the same parity. A pseudo-magic rectangle is an array as above with constant row-sum and column-sum, without the requirement that the elements are in \(\{1,2,\dots,a_1a_2\}\).
A magic rectangle set using integers \(\{1,2,\dots,a_1a_2c\}\) is a set of \(c\) pseudo-magic rectangles, each of order \(a_1\times a_2\), \(2\le a_1\le a_2\), so that all integers from \(1\) through \(a_1a_2\) appear once and only once within the set and all rows in any pseudo-magic rectangle have a constant row-sum \(A_1^{\star}\) and, similarly, all columns in any pseudo-magic rectangle have a constant column-sum, \(A_2^{\star}\). If \(a_1\) or \(a_2\) is odd and \(a_1a_2c\) is even, the magic rectangle set does not exist. \textit{D. Froncek} [AKCE Int. J. Graphs Comb. 10, No. 2, 119--127 (2013; Zbl 1301.05289); Australas. J. Comb. 67, Part 2, 345--351 (2017; Zbl 1375.05234)] has shown that if \(a_1\equiv a_2\mod 2\) and \(a_2\ge 4\) a magic rectangle set exists for every \(c\) and that if \(a_1\), \(a_2\), \(c\) are all positive odd integers so that \(1\le a_1\le a_2\), a magic rectangle exists.
The author shows that magic rectangles and magic rectangles sets are essentially constant block-sum partially balanced incomplete block designs.
A nearly magic rectangle of order \(p\times q\) is an array of order \(p\times q\) that contains integers from the set \(\{1,2,\dots,pq\}\) exactly once, that has constant column-sums and that hat row-sums that differ from each other by no more than \(1\). Such rectangles are then associated with nearly constant block-sum PBIB designs, which minimize the corrected sum of squares.
In the appendix, the author provides some examples of such designs.
Reviewer: Paola Bonacini (Catania)Matrix periods and competition periods of Boolean Toeplitz matriceshttps://zbmath.org/1517.050322023-09-22T14:21:46.120933Z"Cheon, Gi-Sang"https://zbmath.org/authors/?q=ai:cheon.gi-sang"Kang, Bumtle"https://zbmath.org/authors/?q=ai:kang.bumtle"Kim, Suh-Ryung"https://zbmath.org/authors/?q=ai:kim.suh-ryung"Ryu, Homoon"https://zbmath.org/authors/?q=ai:ryu.homoonSummary: In this paper, we study the matrix period and the competition period of Toeplitz matrices over a binary Boolean ring \(\mathbb{B} = \{0, 1 \} \). Given subsets \(S\) and \(T\) of \(\{1, \ldots, n-1 \}\), an \(n \times n\) Toeplitz matrix \(A = T_n \langle S; T \rangle\) is defined to have 1 as the \((i, j)\)-entry if and only if \(j-i \in S\) or \(i - j \in T\). We show that if \(\max S + \min T \leq n\) and \(\min S + \max T \leq n\), then \(A\) has the matrix period \(d / d^\prime\) and the competition period 1 where \(d = \gcd(s + t \mid s \in S, t \in T)\) and \(d^\prime = \gcd(d, \min S)\). Moreover, it is shown that the limit of the matrix sequence \(\{A^m (A^T)^m \}_{m = 1}^\infty\) is a directed sum of matrices of all ones except zero diagonal. In many literatures we see that graph theoretic method can be used to prove strong structural properties about matrices. Likewise, we develop our work from a graph theoretic point of view.Activity from matroids to rooted trees and beyondhttps://zbmath.org/1517.050332023-09-22T14:21:46.120933Z"Flórez, Rigoberto"https://zbmath.org/authors/?q=ai:florez.rigoberto"Forge, David"https://zbmath.org/authors/?q=ai:forge.davidSummary: The interior and exterior activities of bases of a matroid are well-known notions that for instance permit one to define the Tutte polynomial. Recently, we have discovered correspondences between the regions of gainic hyperplane arrangements and colored labeled rooted trees. Here we define a general activity theory that applies in particular to no-broken circuit (NBC) sets and labeled colored trees. The special case of activity \textsf{0} was our motivating case. As a consequence, in a gainic hyperplane arrangement the number of bounded regions is equal to the number of the corresponding colored labeled rooted trees of activity \textsf{0}.The projectivization matroid of a \(q\)-matroidhttps://zbmath.org/1517.050342023-09-22T14:21:46.120933Z"Jany, Benjamin"https://zbmath.org/authors/?q=ai:jany.benjaminSummary: In this paper, we investigate the relation between a \(q\)-matroid and its associated matroid called the projectivization matroid. The latter arises by projectivizing the groundspace of the \(q\)-matroid and considering the projective space as the groundset of the associated matroid on which is defined a rank function compatible with that of the \(q\)-matroid. We show that the projectivization map is a functor from categories of \(q\)-matroids to categories of matroids, which allows us to prove new results about maps of \(q\)-matroids. We furthermore show the characteristic polynomial of a \(q\)-matroid is equal to that of the projectivization matroid. We use this relation to establish a recursive formula for the characteristic polynomial of a \(q\)-matroid in terms of the characteristic polynomial of its minors. Finally we use the projectivization matroid to prove a \(q\)-analogue of the Critical Theorem in terms of \(\mathbb{F}_{q^m}\)-linear rank metric codes and \(q\)-matroids.Sampling planar tanglegrams and pairs of disjoint triangulationshttps://zbmath.org/1517.050352023-09-22T14:21:46.120933Z"Black, Alexander E."https://zbmath.org/authors/?q=ai:black.alexander-e"Liu, Kevin"https://zbmath.org/authors/?q=ai:liu.kevin-fong-rey"McDonough, Alex"https://zbmath.org/authors/?q=ai:mcdonough.alex"Nelson, Garrett"https://zbmath.org/authors/?q=ai:nelson.garrett"Wigal, Michael C."https://zbmath.org/authors/?q=ai:wigal.michael-c"Yin, Mei"https://zbmath.org/authors/?q=ai:yin.mei"Yoo, Youngho"https://zbmath.org/authors/?q=ai:yoo.younghoSummary: A tanglegram consists of two rooted binary trees and a perfect matching between their leaves, and a planar tanglegram is one that admits a layout with no crossings. We show that the problem of generating planar tanglegrams uniformly at random reduces to the corresponding problem for irreducible planar tanglegram layouts, which are known to be in bijection with pairs of disjoint triangulations of a convex polygon. We extend the flip operation on a single triangulation to a flip operation on pairs of disjoint triangulations. Interestingly, the resulting flip graph is both connected and regular, and hence a random walk on this graph converges to the uniform distribution. We also show that the restriction of the flip graph to the pairs with a fixed triangulation in either coordinate is connected, and give diameter bounds that are near optimal. Our results furthermore yield new insight into the flip graph of triangulations of a convex \(n\)-gon with a geometric interpretation on the associahedron.Radius, leaf number, connected domination number and minimum degreehttps://zbmath.org/1517.050362023-09-22T14:21:46.120933Z"Mafuta, P."https://zbmath.org/authors/?q=ai:mafuta.phillip"Mukwembi, S."https://zbmath.org/authors/?q=ai:mukwembi.simon"Munyira, S."https://zbmath.org/authors/?q=ai:munyira.sheunesuSummary: Let \(G\) be a simple, connected graph with minimum degree \(\delta\), radius \(r\) and leaf number \(L(G)\). We prove that
\[
L(G) \geq
\begin{aligned}
\begin{cases}
\frac{2}{3} \left(r - \frac{1}{2}\right) (\delta - 2) + 2 \text{ if } r \equiv 2 \text{ modulo } 3,\\
\frac{2}{3}r (\delta - 2) + 2 \text{ if } r \equiv 0 \text{ modulo } 3,\\
\frac{2}{3} (r - 1) (\delta - 2) + 2 \text{ otherwise}.
\end{cases}
\end{aligned}
\]
We give similar bounds for triangle-free graphs. Infinite families of graphs are constructed to show that all the bounds here are sharp, except the one for \(r \equiv 1\) modulo 3 in the above piecewise inequality. The results bring to literature new lower bounds on the leaf number and new upper bounds on the radius and connected domination number of a graph. Further, the techniques applied in this paper can be used to improve known asymptotically sharp bounds on the radius and diameter to sharp bounds. We consider simple graphs only.A note on a spanning 3-treehttps://zbmath.org/1517.050372023-09-22T14:21:46.120933Z"Tsugaki, Masao"https://zbmath.org/authors/?q=ai:tsugaki.masaoSummary: A tree \(T\) is called a \(k\)-tree, if the maximum degree of \(T\) is at most \(k\). In this paper, we prove that if \(G\) is an \(n\)-connected graph with independence number at most \(n + m + 1 (n \geq 1,n \geq m \geq 0)\), then \(G\) has a spanning 3-tree \(T\) with at most \(m\) vertices of degree 3.Wiener index and addressing of some finite graphshttps://zbmath.org/1517.050382023-09-22T14:21:46.120933Z"Gholamnia Taleshani, Mona"https://zbmath.org/authors/?q=ai:gholamnia-taleshani.mona"Abbasi, Ahmad"https://zbmath.org/authors/?q=ai:abbasi.ahmadSummary: An addressing of length \(t\) of a graph \(G\) is an assignment of \(t\)-tuples with entries in \(\{0,a,b\}\), called \textit{addresses}, to the vertices of \(G\) such that the distance between any two vertices can be determined from their addresses. Let \(Z(R)\) be the set of zero-divisors of a commutative ring \(R\). In this article, we investigate the adjacency matrix of \(Z(\Gamma(\mathbb{Z}_{p^{\alpha}q^{\beta}}))\), a simple graph whose vertex set is \(Z(\mathbb{Z}_{p^{\alpha}q^{\beta}})\) and two distinct vertices \(x\) and \(y\) are adjacent if and only if \(x+y \in Z(\mathbb{Z}_{p^{\alpha}q^{\beta}})\), where \(p<q\) are distinct primes greater than two and \(\alpha\), \(\beta\) are positive integers. Moreover, we estimate addressing of \((Z(\Gamma(\mathbb{Z}_{p^{\alpha}q^{\beta}}))\) and obtain Wiener and Zagreb indices and some energies of it.Maximum and minimum values of inverse degree and forgotten indices on the class of all unicyclic graphshttps://zbmath.org/1517.050392023-09-22T14:21:46.120933Z"Manian, Mohammad Ali"https://zbmath.org/authors/?q=ai:manian.mohammad-ali"Heidarian, Shahram"https://zbmath.org/authors/?q=ai:heidarian.shahram"Khaksar Haghani, Farhad"https://zbmath.org/authors/?q=ai:haghani.farhad-khaksarSummary: For a connected simple graph \(G\), the inverse degree index and forgotten index are defined as \(\mathrm{ID}(G) = \sum_{u \in V(G)} \frac{1}{d_u}\) and \(F(G) = \sum_{uv \in E(G)} [d_u^2 + d_v^2]\) respectively, where \(d_u\) denotes the degree of vertex \(u\) in \(G\). In this article, we use some graph transformations and introduce a method for determining the extremal values of inverse degree index and forgotten index on the class of unicyclic graphs.Maximizing the Mostar index for bipartite graphs and split graphshttps://zbmath.org/1517.050402023-09-22T14:21:46.120933Z"Miklavič, Štefko"https://zbmath.org/authors/?q=ai:miklavic.stefko"Pardey, Johannes"https://zbmath.org/authors/?q=ai:pardey.johannes"Rautenbach, Dieter"https://zbmath.org/authors/?q=ai:rautenbach.dieter"Werner, Florian"https://zbmath.org/authors/?q=ai:werner.florianSummary: \textit{T. Došlić} et al. [J. Math. Chem. 56, No. 10, 2995--3013 (2018; Zbl 1406.92750)] defined the Mostar index of a graph \(G\) as \(\sum\limits_{uv\in E(G)}|n_G(u,v)-n_G(v,u)|\), where, for an edge \(uv\) of \(G\), the term \(n_G(u,v)\) denotes the number of vertices of \(G\) that have a smaller distance in \(G\) to \(u\) than to \(v\). Contributing to conjectures posed by Došlić et al. [loc. cit.], we show that the Mostar index of bipartite graphs of order \(n\) is at most \(\frac{\sqrt{3}}{18}n^3\), and that the Mostar index of split graphs of order \(n\) is at most \(\frac{4}{27}n^3\).Random cubic planar graphs converge to the Brownian spherehttps://zbmath.org/1517.050412023-09-22T14:21:46.120933Z"Albenque, Marie"https://zbmath.org/authors/?q=ai:albenque.marie"Fusy, Éric"https://zbmath.org/authors/?q=ai:fusy.eric"Lehéricy, Thomas"https://zbmath.org/authors/?q=ai:lehericy.thomasSummary: In this paper, the scaling limit of random connected cubic planar graphs (respectively multigraphs) is shown to be the Brownian sphere.
The proof consists in essentially two main steps. First, thanks to the known decomposition of cubic planar graphs into their 3-connected components, the metric structure of a random cubic planar graph is shown to be well approximated by its unique 3-connected component of linear size, with modified distances.
Then, Whitney's theorem ensures that a 3-connected cubic planar graph is the dual of a simple triangulation, for which it is known that the scaling limit is the Brownian sphere. \textit{N. Curien} and \textit{J.-F. Le Gall} [Ann. Sci. Éc. Norm. Supér. (4) 52, No. 3, 631--701 (2019; Zbl 1429.05188)] have recently developed a framework to study the modification of distances in general triangulations and in their dual. By extending this framework to simple triangulations, it is shown that 3-connected cubic planar graphs with modified distances converge jointly with their dual triangulation to the Brownian sphere.The crossing numbers of join products of eight graphs of order six with paths and cycleshttps://zbmath.org/1517.050422023-09-22T14:21:46.120933Z"Staš, M."https://zbmath.org/authors/?q=ai:stas.michalSummary: The crossing number \(\text{cr}(G)\) of a graph \(G\) is the minimum number of edge crossings over all drawings of \(G\) in the plane. The main aim of this paper is to give the crossing numbers of the join products of eight graphs on six vertices with paths and cycles on \(n\) vertices. The proofs are done with the help of several well-known auxiliary statements, the idea of which is extended by a suitable classification of subgraphs that do not cross the edges of the examined graphs.The localization number and metric dimension of graphs of diameter 2https://zbmath.org/1517.050432023-09-22T14:21:46.120933Z"Bonato, Anthony"https://zbmath.org/authors/?q=ai:bonato.anthony"Huggan, Melissa"https://zbmath.org/authors/?q=ai:huggan.melissa-a"Marbach, Trent"https://zbmath.org/authors/?q=ai:marbach.trent-gregorySummary: We consider the localization number and metric dimension of certain graphs of diameter 2, focusing on families of Kneser graphs and graphs without 4-cycles. For the Kneser graphs with a diameter of 2, we find upper and lower bounds for the localization number and metric dimension, and in many cases these parameters differ only by an additive constant. Our results on the metric dimension of Kneser graphs improve on earlier ones, yielding exact values in infinitely many cases. We determine bounds on the localization number and metric dimension of Moore graphs of diameter 2 and polarity graphs.Proximity, remoteness and maximum degree in graphshttps://zbmath.org/1517.050442023-09-22T14:21:46.120933Z"Dankelmann, Peter"https://zbmath.org/authors/?q=ai:dankelmann.peter"Mafunda, Sonwabile"https://zbmath.org/authors/?q=ai:mafunda.sonwabile"Mallu, Sufiyan"https://zbmath.org/authors/?q=ai:mallu.sufiyanLet \(G\) be a graph. The total distance of \(v\in V(G)\) is \(\sigma(v)=\sum_{u\in V(G)}d(v,u)\), where \(d(u,v)\) is the usual shortest path distance in \(G\). The average shortest path distance is \(\overline{\sigma}(v)=\frac{1}{n-1}\sigma(v)\). The proximity of \(G\) is \(\pi(G)=\max_{v\in V(G)}\{\overline{\sigma}(v)\}\) and remoteness of \(G\) is \(\rho(G)=\min_{v\in V(G)}\{\overline{\sigma}(v)\}\).
This contribution improves previous upper bounds of \(\pi(G)\) and \(\rho(G)\) with respect to the order \(|V(G)|\) of \(G\) and minimum degree \(\delta(G)\) of \(G\) with addition of the maximum degree \(\Delta(G)\) of \(G\).
Reviewer: Iztok Peterin (Maribor)Distance problems within Helly graphs and \(k\)-Helly graphshttps://zbmath.org/1517.050452023-09-22T14:21:46.120933Z"Ducoffe, Guillaume"https://zbmath.org/authors/?q=ai:ducoffe.guillaumeSummary: The ball hypergraph of a graph \(G\) is the family of balls of all possible centers and radii in \(G\). Balls in a subfamily are \(k\)-wise intersecting if the intersection of any \(k\) balls in the subfamily is always nonempty. The Helly number of a ball hypergraph is the least integer \(k\) greater than one such that every subfamily of \(k\)-wise intersecting balls has a nonempty common intersection. A graph is \(k\)-Helly (or Helly, if \(k = 2\)) if its ball hypergraph has a Helly number at most \(k\). We prove that a central vertex and all the medians in an \(n\)-vertex \(m\)-edge Helly graph can be computed w.h.p. in \(\widetilde{\mathcal{O}}(m \sqrt{ n})\) time. Both results extend to a broader setting where we assign a nonnegative cost to the vertices. For any fixed \(k\), we also present an \(\widetilde{\mathcal{O}}(m \sqrt{ k n})\)-time randomized algorithm for radius computation within \(k\)-Helly graphs. If we relax the definition of Helly number (for what is sometimes called an ``almost Helly-type'' property in the literature), then our approach leads to an approximation algorithm for computing the radius with an additive one-sided error of at most some constant.Edge general position sets in Fibonacci and Lucas cubeshttps://zbmath.org/1517.050462023-09-22T14:21:46.120933Z"Klavžar, Sandi"https://zbmath.org/authors/?q=ai:klavzar.sandi"Tan, Elif"https://zbmath.org/authors/?q=ai:tan.elifSummary: A set of edges \(X\subseteq E(G)\) of a graph \(G\) is an edge general position set if no three edges from \(X\) lie on a common shortest path in \(G\). The cardinality of a largest edge general position set of \(G\) is the edge general position number of \(G\). In this paper, edge general position sets are investigated in partial cubes. In particular, it is proved that the union of two largest \(\Theta\)-classes of a Fibonacci cube or a Lucas cube is a maximal edge general position set.On the edge chromatic vertex stability number of graphshttps://zbmath.org/1517.050472023-09-22T14:21:46.120933Z"Alikhani, Saeid"https://zbmath.org/authors/?q=ai:alikhani.saeid"Piri, Mohammad R."https://zbmath.org/authors/?q=ai:piri.mohammad-rezaSummary: For an arbitrary invariant \(\rho(G)\) of a graph \(G\), the \(\rho\)-vertex stability number \(vs_{\rho}(G)\) is the minimum number of vertices of \(G\) whose removal results in a graph \(H \subseteq G\) with \(\rho(H) \neq \rho(G)\) or with \(E(H) = \emptyset\). In this paper, first we give some general lower and upper bounds for the \(\rho\)-vertex stability number, and then study the edge chromatic vertex stability number of graphs, \(vs_{\chi'}(G)\) where \(\chi' = \chi' (G)\) is edge chromatic number (chromatic index) of \(G\). We prove some general results for this parameter and determine \(vs_{\chi'}(G)\) for specific classes of graphs.Rainbow mean colorings of some classes of graphshttps://zbmath.org/1517.050482023-09-22T14:21:46.120933Z"Anantharaman, S."https://zbmath.org/authors/?q=ai:anantharaman.sreeraman|anantharaman.siva"Sampathkumar, R."https://zbmath.org/authors/?q=ai:sampathkumar.rathinasamy"Sivakaran, T."https://zbmath.org/authors/?q=ai:sivakaran.tSummary: For an edge-coloring \(c\) of a connected simple graph \(G\) with positive integers, where adjacent edges may be colored the same, the chromatic mean of a vertex \(v\) of \(G\) is the average of the colors of the edges incident with \(v\). Only those edge-colorings \(c\) for which the chromatic mean of every vertex is a positive integer are considered. If distinct vertices have distinct chromatic means, then \(c\) is a rainbow mean coloring of \(G\). The maximum chromatic mean of vertices of \(G\) in a rainbow mean coloring \(c\) of \(G\) is the rainbow mean index of \(c\), while the rainbow mean index of \(G\) is the minimum rainbow mean index among all rainbow mean colorings of \(G\). In this paper, we determine the rainbow mean index of several classes of connected simple graphs, namely, a connected regular graph containing a \(P_4\)-factor, a connected regular graph containing a vertex \(v\) such that the subgraph \(G-v\) has a \(P_4\)-factor, a connected odd regular graph of order \(n\) with \(n\equiv 2\pmod 4\) and containing a vertex \(v\) such that the subgraph \(G-v\) has a factor \(H\) in which one component of \(H\) is \(P_5\) and all other components of \(H\) are isomorphic to \(P_4\), lexicographic product of a connected regular graph \(G\) containing a 1-factor and a connected graph \(H\), some caterpillars, the complete tripartite graph \(K_{1,1,n}\), the friendship graph, the fan, and the wheel.Distributed algorithms, the Lovász local lemma, and descriptive combinatoricshttps://zbmath.org/1517.050492023-09-22T14:21:46.120933Z"Bernshteyn, Anton"https://zbmath.org/authors/?q=ai:bernshteyn.antonSummary: In this paper we consider coloring problems on graphs and other combinatorial structures on standard Borel spaces. Our goal is to obtain sufficient conditions under which such colorings can be made well-behaved in the sense of topology or measure. To this end, we show that such well-behaved colorings can be produced using certain powerful techniques from finite combinatorics and computer science. First, we prove that efficient distributed coloring algorithms (on finite graphs) yield well-behaved colorings of Borel graphs of bounded degree; roughly speaking, deterministic algorithms produce Borel colorings, while randomized algorithms give measurable and Baire-measurable colorings. Second, we establish measurable and Baire-measurable versions of the Symmetric Lovász Local Lemma (under the assumption \(\mathsf{p}(\mathsf{d}+1)^8 \leqslant 2^{-15} \), which is stronger than the standard LLL assumption \(\mathsf{p}(\mathsf{d}+ 1) \leqslant e^{-1}\) but still sufficient for many applications). From these general results, we derive a number of consequences in descriptive combinatorics and ergodic theory.Nonrepetitive list colorings of the integershttps://zbmath.org/1517.050502023-09-22T14:21:46.120933Z"Bosek, Bartłomiej"https://zbmath.org/authors/?q=ai:bosek.bartlomiej"Grytczuk, Jarosław"https://zbmath.org/authors/?q=ai:grytczuk.jaroslaw"Nayar, Barbara"https://zbmath.org/authors/?q=ai:nayar.barbara"Zaleski, Bartosz"https://zbmath.org/authors/?q=ai:zaleski.bartoszSummary: A coloring of the integers is nonrepetitive if no two adjacent intervals have the same color sequence. A beautiful theorem of Thue asserts that there exists a nonrepetitive coloring of \(\mathbb{N}\) using only three colors. We obtain some generalizations of this result in which the adjacency of intervals is specified by more general graphs. We focus on the list variant of the problem, in which every integer gets a color from its own set of colors. For instance, we prove that there exists a coloring of \(\mathbb{N}\) from arbitrary lists of size 8, such that the following property holds for every \(n\geq 1\): among any \(2^n\) consecutively adjacent intervals, each of length \(n\), no two have the same color sequence. Another result is related to the possible extension of the famous Dejean's conjecture to the list setting. It asserts that for every \(k\geq 1\), there is a coloring of \(\mathbb{N}\) from lists of size \(k+2\sqrt{k}\), such that no two among any \(k\) consecutively adjacent intervals have the same color sequence.A note on the four color theoremhttps://zbmath.org/1517.050512023-09-22T14:21:46.120933Z"Canizales, Jennifer"https://zbmath.org/authors/?q=ai:canizales.jennifer"Chahal, Jasbir S."https://zbmath.org/authors/?q=ai:chahal.jasbir-singhThe four-color problem has been a major driving force to reach on various aspects of graph coloring. In spite of extensive work by several leading experts for more than 175 years, a computer-free proof of the four-color theorem is not available until today. One possible approach for a computer-free proof of the four-color theorem is to show that the chromatic polynomial of any planar graph evaluated at 4 is nonzero. In this paper, the authors prove that for the family of 2 interlocking wheels, the chromatic polynomial evaluated at 4 is nonzero. The problem for \(k\) interlocking wheels where \(k>3\) is left open.
Reviewer: S. Arumugam (Krishnankoil)Integer colorings with forbidden rainbow sumshttps://zbmath.org/1517.050522023-09-22T14:21:46.120933Z"Cheng, Yangyang"https://zbmath.org/authors/?q=ai:cheng.yangyang"Jing, Yifan"https://zbmath.org/authors/?q=ai:jing.yifan"Li, Lina"https://zbmath.org/authors/?q=ai:li.lina.2"Wang, Guanghui"https://zbmath.org/authors/?q=ai:wang.guanghui"Zhou, Wenling"https://zbmath.org/authors/?q=ai:zhou.wenlingSummary: For a set of positive integers \(A\subseteq [n]\), an \(r\)-coloring of \(A\) is rainbow sum-free if it contains no rainbow Schur triple. In this paper we initiate the study of the rainbow Erdős-Rothschild problem in the context of sum-free sets, which asks for the subsets of \([n]\) with the maximum number of rainbow sum-free \(r\)-colorings. We show that for \(r=3\), the interval \([n]\) is optimal, while for \(r\geq 8\), the set \([\lfloor n/2\rfloor,n]\) is optimal. We also prove a stability theorem for \(r\geq 4\). The proofs rely on the hypergraph container method, and some ad-hoc stability analysis.Minimum gradation in greyscales of graphshttps://zbmath.org/1517.050532023-09-22T14:21:46.120933Z"de Castro, Natalia"https://zbmath.org/authors/?q=ai:de-castro.natalia"Garrido-Vizuete, María A."https://zbmath.org/authors/?q=ai:garrido-vizuete.maria-a"Robles, Rafael"https://zbmath.org/authors/?q=ai:robles.rafael"Villar-Liñán, María Trinidad"https://zbmath.org/authors/?q=ai:villar-linan.m-tSummary: In this paper we present the notion of greyscale of a graph as a colouring of its vertices that uses colours from the real interval [0,1]. Any greyscale induces another colouring by assigning to each edge the non-negative difference between the colours of its vertices. These edge colours are ordered in lexicographical decreasing ordering and give rise to a new element of the graph: the gradation vector. We introduce the notion of minimum gradation vector as a new invariant for the graph and give polynomial algorithms to obtain it. These algorithms also output all greyscales that produce the minimum gradation vector. This way we tackle and solve a novel vectorial optimization problem in graphs that may generate more satisfactory solutions than those generated by known scalar optimization approaches.The edge colorings of \(K_5\)-minor free graphshttps://zbmath.org/1517.050542023-09-22T14:21:46.120933Z"Feng, Jieru"https://zbmath.org/authors/?q=ai:feng.jieru"Gao, Yuping"https://zbmath.org/authors/?q=ai:gao.yuping"Wu, Jianliang"https://zbmath.org/authors/?q=ai:wu.jian-liangAn edge \(k\)-coloring of a graph \(G\) is an assignment of \(k\) colors to the edges of \(G\) such that no two adjacent edges receive the same color. Vizing proved that every planar graph with maximum degree \(\Delta\geq 8\) is edge \(\Delta\)-colorable [\textit{V. G. Vizing}, Diskret. Analiz, Novosibirsk 5, 9--17 (1965; Zbl 0171.44902)]. It is also proved, by \textit{D. P. Sanders} and \textit{Y. Zhao} [J. Comb. Theory, Ser. B 83, No. 2, 201--212 (2001; Zbl 1024.05031)], and independently by \textit{L. Zhang} [Graphs Comb. 16, No. 4, 467--495 (2000; Zbl 0988.05042)] that every planar graph with maximum degree \(\Delta = 7\) is edge \(\Delta\)-colorable. In this paper, the authors extend these results by proving that every \(K_5\)-minor free graph with maximum degree \(\Delta \geq 7\) is edge \(\Delta\)-colorable.
Reviewer: Juan José Montellano Ballesteros (Ciudad de México)A generalization of properly colored paths and cycles in edge-colored graphshttps://zbmath.org/1517.050552023-09-22T14:21:46.120933Z"Galeana-Sánchez, Hortensia"https://zbmath.org/authors/?q=ai:galeana-sanchez.hortensia"Hernández-Lorenzana, Felipe"https://zbmath.org/authors/?q=ai:hernandez-lorenzana.felipe"Sánchez-López, Rocío"https://zbmath.org/authors/?q=ai:sanchez-lopez.rocioSummary: A properly colored walk, is a walk where every two consecutive edges have different colors, including the first and last edges in the case when the walk is closed. Properly colored walks have shown to be an effective way to model certain real applications. In view of this, it is natural to ask about the existence of properly colored walks with restrictions on the transitions of colors allowed in the edges of a graph. Let \(H\) be a graph, possibly with loops, and \(G\) a graph. We will say that \(G\) is \(H\)-colored iff there exists a function \(c : E(G) \longrightarrow V(H)\). A path \(( v_0, v_1, \dots, v_k)\) in \(G\) is an \(H\)-path whenever \((c( v_0 v_1), \dots, c( v_{k - 1} v_k))\) is a walk in \(H\); in particular, a cycle \(( v_0, v_1, \dots, v_k, v_0)\) is an \(H\)-cycle iff \((c( v_0 v_1), c( v_1 v_2), \dots, c( v_{k - 1} v_k), c( v_k v_0), c( v_0 v_1))\) is a walk in \(H\). Hence, the graph \(H\) determines what color transitions are allowed in a walk. An interesting application of \(H\)-walks states as follows: suppose that it is required to send a message through a hostile environment (network), in which the edges are colored with its transmission code; in order to make more difficult the decoding task, it is necessary to avoid not only monochromatic transitions, but some specific color transitions that make easier that task. Some authors have studied the existence of \(H\)-cycles in an \(H\)-colored graph, saying nothing about its length. In this paper, we give conditions that imply the existence of \(H\)-paths and \(H\)-cycles of certain length, in an \(H\)-colored graph with a given structure. As a consequence of the main results, we obtain known theorems in the theory of properly colored walks.Total dominator chromatic number of Kneser graphshttps://zbmath.org/1517.050562023-09-22T14:21:46.120933Z"Jalilolghadr, Parvin"https://zbmath.org/authors/?q=ai:jalilolghadr.parvin"Behtoei, Ali"https://zbmath.org/authors/?q=ai:behtoei.aliSummary: Decomposition into special substructures inheriting significant properties is an important method for the investigation of some mathematical structures. A \textit{total dominator coloring} (briefly, a TDC) of a graph \(G\) is a proper coloring (i.e. a partition of the vertex set \(V(G)\) into independent subsets named color classes) in which each vertex of the graph is adjacent to all of vertices of some color class. The \textit{total dominator chromatic number} \(\chi_{\mathrm{td}}(G)\) of \(G\) is the minimum number of color classes in a TDC of \(G\). In this paper among some other results and by using the existence of Steiner triple systems, we determine the total dominator chromatic number of the Kneser graph \(\mathrm{KG}(n,2)\) for each \(n \geq 5\).Algorithms for the rainbow vertex coloring problem on graph classeshttps://zbmath.org/1517.050572023-09-22T14:21:46.120933Z"Lima, Paloma T."https://zbmath.org/authors/?q=ai:lima.paloma-t"van Leeuwen, Erik Jan"https://zbmath.org/authors/?q=ai:van-leeuwen.erik-jan"van der Wegen, Marieke"https://zbmath.org/authors/?q=ai:van-der-wegen.mariekeSummary: Given a vertex-colored graph, we say a path is a rainbow vertex path if all its internal vertices have distinct colors. The graph is rainbow vertex-connected if there is a rainbow vertex path between every pair of its vertices. In the \textsc{Rainbow Vertex Coloring (RVC)} problem we want to decide whether the vertices of a given graph can be colored with at most \(k\) colors so that the graph becomes rainbow vertex-connected. This problem is known to be NP-complete even in very restricted scenarios, and very few efficient algorithms are known for it. In this work, we give polynomial-time algorithms for RVC on permutation graphs, powers of trees and split strongly chordal graphs. The algorithm for the latter class also works for the strong variant of the problem, where the rainbow vertex paths between each vertex pair must be shortest paths. We complement the polynomial-time solvability results for split strongly chordal graphs by showing that, for any fixed \(p \geq 3\) both variants of the problem become NP-complete when restricted to split \(( S_3, \ldots, S_p)\)-free graphs, where \(S_q\) denotes the \(q\)-sun graph.Role colouring graphs in hereditary classeshttps://zbmath.org/1517.050582023-09-22T14:21:46.120933Z"Purcell, Christopher"https://zbmath.org/authors/?q=ai:purcell.christopher-j"Rombach, Puck"https://zbmath.org/authors/?q=ai:rombach.puckSummary: A locally surjective homomorphism from a graph \(G\) to a graph \(H\) is called an \(H\)-role colouring of \(G\). Deciding the existence of such a colouring with \(| H | = k\) is known to be NP-hard even under substantial restrictions on the input graph \(G\). We study the family of hereditary classes for which the problem can be solved efficiently. Our main result is the first boundary class for this problem; that is, a minimal obstruction to solving the problem efficiently in a finitely defined hereditary class. We also give the first boundary class for the related \(k\)-coupon colouring problem, in which \(H\) is a complete graph with each vertex incident to a loop, when \(k\) is a prime power. As additional results, we show that 2-role colouring \(2 K_2\)-free graphs and 2-coupon colouring cographs can be done in linear time.Acyclic edge coloring conjecture is true on planar graphs without intersecting triangleshttps://zbmath.org/1517.050592023-09-22T14:21:46.120933Z"Shu, Qiaojun"https://zbmath.org/authors/?q=ai:shu.qiaojun"Chen, Yong"https://zbmath.org/authors/?q=ai:chen.yong.3"Han, Shuguang"https://zbmath.org/authors/?q=ai:han.shuguang"Lin, Guohui"https://zbmath.org/authors/?q=ai:lin.guohui"Miyano, Eiji"https://zbmath.org/authors/?q=ai:miyano.eiji"Zhang, An"https://zbmath.org/authors/?q=ai:zhang.anSummary: An acyclic edge coloring of a graph \(G\) is a proper edge coloring such that no bichromatic cycles are produced. The acyclic edge coloring conjecture by \textit{I. Fiamcik} [Math. Slovaca 28, 139--145 (1978; Zbl 0388.05015)] and \textit{N. Alon} et al. [J. Graph Theory 37, No. 3, 157--167 (2001; Zbl 0996.05050)] states that every simple graph with maximum degree \(\Delta\) is acyclically edge \((\Delta + 2)\)-colorable. Despite many milestones, the conjecture remains open even for planar graphs. In this paper, we confirm affirmatively the conjecture on planar graphs without intersecting triangles. We do so by first showing, by discharging methods, that every planar graph without intersecting triangles must have at least one of the six specified groups of local structures, and then proving the conjecture by recoloring certain edges in each such local structure and by induction on the number of edges in the graph.Acyclic edge coloring conjecture is true on planar graphs without intersecting triangleshttps://zbmath.org/1517.050602023-09-22T14:21:46.120933Z"Shu, Qiaojun"https://zbmath.org/authors/?q=ai:shu.qiaojun"Chen, Yong"https://zbmath.org/authors/?q=ai:chen.yong.3"Han, Shuguang"https://zbmath.org/authors/?q=ai:han.shuguang"Lin, Guohui"https://zbmath.org/authors/?q=ai:lin.guohui"Miyano, Eiji"https://zbmath.org/authors/?q=ai:miyano.eiji"Zhang, An"https://zbmath.org/authors/?q=ai:zhang.anSummary: An acyclic edge coloring of a graph \(G\) is a proper edge coloring such that no bichromatic cycles are produced. The acyclic edge coloring conjecture by \textit{I. Fiamcik} [Math. Slovaca 28, 139--145 (1978; Zbl 0388.05015)] and \textit{N. Alon} et al. [J. Graph Theory 37, No. 3, 157--167 (2001; Zbl 0996.05050)] states that every simple graph with maximum degree \(\varDelta\) is acyclically edge \((\varDelta + 2)\)-colorable. Despite many milestones, the conjecture is still unknown true or not even for planar graphs. In this paper, we first show by discharging methods that every planar graph without intersecting triangles must have one of the six specified groups of local structures; then by induction on the number of edges we confirm affirmatively the conjecture on planar graphs without intersecting triangles.
For the entire collection see [Zbl 1502.68022].Equitable total coloring of tensor product of graphshttps://zbmath.org/1517.050612023-09-22T14:21:46.120933Z"Vivik, J. Veninstine"https://zbmath.org/authors/?q=ai:veninstine-vivik.j"Girija, G."https://zbmath.org/authors/?q=ai:girija.gSummary: The equitable total coloring of a graph \(G\) is a proper total coloring, where the number of elements (vertices and edges) in any two color classes differ by at most one and the minimum number of required colors required is its equitable total chromatic number. In this paper, we have discussed the equitable total chromatic number of \(S^0_n\), \(P_m\otimes S^0_n\), \(S^0_m\otimes S^0_n\) and \(K_{1,m}\otimes K_{1,n}\).On the chromatic number of \(( P_5, \mathrm{dart})\)-free graphshttps://zbmath.org/1517.050622023-09-22T14:21:46.120933Z"Xu, Weilun"https://zbmath.org/authors/?q=ai:xu.weilun"Zhang, Xia"https://zbmath.org/authors/?q=ai:zhang.xiaSummary: For a graph \(G\), \(\omega(G)\), \(\chi(G)\) represent the clique number and the chromatic number of \(G\), respectively. A hereditary family \(\mathcal{G}\) of graphs is called \(\chi\)-bounded with \(\chi\)-binding function \(f\) if \(\chi (G)\leq f (\omega (G))\) for all \(G\in\mathcal{G}\). A result of Schiermeyer shows that the class of \((P_5,\mathrm{dart})\)-free graphs has a \(\chi\)-binding function \(f (\omega)= \omega^2\) [\textit{L. Esperet} et al., Discrete Math. 313, No. 6, 743--754 (2013; Zbl 1260.05056)]. In this paper, we prove that the class of \((P_5,\mathrm{dart})\)-free graphs has a \(\chi\)-binding function \(f (\omega)=\frac{3}{4} \omega^2\). For the class of \((P_5, C_5,\mathrm{dart})\)-free graphs, we give a \(\chi\)-binding function \(f(\omega)=\binom{\omega +1}{2}\).On the proper orientation number of chordal graphshttps://zbmath.org/1517.050632023-09-22T14:21:46.120933Z"Araujo, J."https://zbmath.org/authors/?q=ai:araujo.julio-cesar-silva"Cezar, A."https://zbmath.org/authors/?q=ai:cezar.alexandre"Lima, C. V. G. C."https://zbmath.org/authors/?q=ai:lima.carlos-vinicius-gomes-costa"dos Santos, V. F."https://zbmath.org/authors/?q=ai:dos-santos.vinicius-fernandes"Silva, A."https://zbmath.org/authors/?q=ai:silva.ana-r-s|silva.andre-luis-ferreira-da|silva.ariosto-s|silva.alexandru|silva.a-de-a-e|silva.a-pedro-duarte|silva.ana-carolina|silva.ana-maria-f|silva.anderson-t|silva.adriano-w|silva.anibal|silva.alexandre-m-f|silva.arlenes-s|silva.adilson|silva.analia|silva.arlaine-a|silva.alberto|silva.alessandro|silva.anne|silva.andres|silva.ana-t-c|silva.ana-l|silva.antonio-m-p|silva.a-i|silva.a-christian|silva.augusto|silva.adriano-da|silva.adriana-r|silva.alexandra|silva.alonso|silva.andre-c|silva.antonio-m-m|silva.aline-p|silva.alisson-r|silva.andrea-r-d|silva.a-f-c|silva.aristofanes-correa|silva.aldy-f|silva.allyson|silva.antonio-kelson|silva.arlindo|silva.allessandro|silva.adao|silva.apollo-v|silva.antonio-n-jun|silva.andreia-f|silva.anabela-sousaSummary: An orientation \(D\) of a graph \(G = (V, E)\) is a digraph obtained from \(G\) by replacing each edge by exactly one of the two possible arcs with the same end vertices. For each \(v \in V(G)\), the indegree of \(v\) in \(D\), denoted by \(d_D^-(v)\), is the number of arcs with head \(v\) in \(D\). An orientation \(D\) of \(G\) is proper if \(d_D^-(u) \neq d_D^-(v)\), for all \(u v \in E(G)\). An orientation with maximum indegree at most \(k\) is called a \(k\)-orientation. The proper orientation number of \(G\), denoted by \(\overrightarrow{\chi}(G)\), is the minimum integer \(k\) such that \(G\) admits a proper \(k\)-orientation. We prove that determining whether \(\overrightarrow{\chi}(G) \leq k\) is \(\mathsf{NP}\)-complete for chordal graphs of bounded diameter, but can be solved in linear-time in the subclass of quasi-threshold graphs. When parameterizing by \(k\), we argue that this problem is \textsf{FPT} for chordal graphs and argue that no polynomial kernel exists, unless \(\mathsf{NP} \subseteq \mathsf{coNP} / \mathsf{poly} \). We present a better kernel to the subclass of split graphs and a linear kernel to the class of cobipartite graphs.
Concerning bounds, we first prove that if \(G\) is split, then \(\overrightarrow{\chi}(G) \leq 2 \omega(G) - 2\) and that if \(G\) is a \(k\)-uniform block graph, then \(\overrightarrow{\chi}(G) \leq 3 k - 2\). These bounds are tight. We also present new families of trees having proper orientation number at most 2 and at most 3. Actually, we prove a general bound stating that any graph \(G\) having no adjacent vertices of degree at least \(c + 1\) has proper orientation number at most \(c\). This implies new classes of (outer)planar graphs with bounded proper orientation number. We also prove that maximal outerplanar graphs \(G\) whose weak-dual is a path satisfy \(\overrightarrow{\chi}(G) \leq 13\). Finally, we present simple bounds to the classes of chordal claw-free graphs and cographs.A family of exact 2-extensions of tournamentshttps://zbmath.org/1517.050642023-09-22T14:21:46.120933Z"Dolgov, A. A."https://zbmath.org/authors/?q=ai:dolgov.a-aSummary: The family of tournaments which have exact 1- and 2-extensions but haven't exact 3-extension is introduced. It is the only known family of graphs with such a property, and it is the fourth family of graphs which have exact \(k\)-extension for \(k>1\).Competition periods of multipartite tournamentshttps://zbmath.org/1517.050652023-09-22T14:21:46.120933Z"Jung, Ji-Hwan"https://zbmath.org/authors/?q=ai:jung.ji-hwan"Kim, Suh-Ryung"https://zbmath.org/authors/?q=ai:kim.suh-ryung"Yoon, Hyesun"https://zbmath.org/authors/?q=ai:yoon.hyesunA \(k\)-partite tournament is an orientation of a complete \(k\)-partite graph for a positive integer \(k\). The adjacency matrix of a \(k\)-partite tournament for \(k\geq 2\) is represented as a \((0, 1)\) Boolean block matrix. The paper is to show that the competition period of a \(k\)-partite tournament is at most three for any integer \(k\geq 2\). Further, if \(k=2\), then the competition period is at most two. The authors also show that the competition period of a multipartite tournament is at most 3, and suggest computing the competition index of a multipartite tournament, where the competition index of a bipartite tournament was given early in the literature.
Reviewer: Wai-Kai Chen (Fremont)Stability regions of systems with compatibilities and ubiquitous measures on graphshttps://zbmath.org/1517.050662023-09-22T14:21:46.120933Z"Begeot, Jocelyn"https://zbmath.org/authors/?q=ai:begeot.jocelyn"Marcovici, Irène"https://zbmath.org/authors/?q=ai:marcovici.irene"Moyal, Pascal"https://zbmath.org/authors/?q=ai:moyal.pascalSummary: This paper addresses the ubiquity of remarkable measures on graphs and their applications. In many queueing systems, it is necessary to take into account the compatibility constraints between users, or between supplies and demands, and so on. The stability region of such systems can then be seen as a set of measures on graphs, where the measures under consideration represent the arrival flows to the various classes of users, supplies, demands, etc., and the graph represents the compatibilities between those classes. In this paper, we show that these `stabilizing' measures can always be easily constructed as a simple function of a family of weights on the edges of the graph. Second, we show that the latter measures always coincide with invariant measures of random walks on the graph under consideration. Some arguments in the proofs rely on the so-called matching rates of specific stochastic matching models. As a by-product of these arguments, we show that, in several cases, the matching rates are independent of the matching policy, that is, the rule for choosing a match between various compatible elements.Inverse problems for discrete heat equations and random walks for a class of graphshttps://zbmath.org/1517.050672023-09-22T14:21:46.120933Z"Blåsten, Emilia"https://zbmath.org/authors/?q=ai:blasten.emilia-l-k"Isozaki, Hiroshi"https://zbmath.org/authors/?q=ai:isozaki.hiroshi"Lassas, Matti"https://zbmath.org/authors/?q=ai:lassas.matti-j"Lu, Jinpeng"https://zbmath.org/authors/?q=ai:lu.jinpengSummary: We study the inverse problem of determining a finite weighted graph \((X,E)\) from the source-to-solution map on a vertex subset \(B\subset X\) for heat equations on graphs, where the time variable can be either discrete or continuous. We prove that this problem is equivalent to the discrete version of the inverse interior spectral problem, provided that there does not exist a nonzero eigenfunction of the weighted graph Laplacian vanishing identically on \(B\). In particular, we consider inverse problems for discrete-time random walks on finite graphs. We show that under a novel geometric condition (called the Two-Points Condition), the graph structure and the transition matrix of the random walk can be uniquely recovered from the distributions of the first passing times on \(B\), or from the observation on \(B\) of one realization of the random walk.The characterization of the minimal weighted acyclic graphshttps://zbmath.org/1517.050682023-09-22T14:21:46.120933Z"Du, Zhibin"https://zbmath.org/authors/?q=ai:du.zhibin"da Fonseca, Carlos M."https://zbmath.org/authors/?q=ai:da-fonseca.carlos-martinsSummary: In this paper, we completely characterize all the trees on \(n\) vertices with diameter \(d\) for which there is a symmetric matrix with nullity \(n-d\) and \(n-d-1\), respectively. These characterizations cover all recent results proved for the standard \(0-1\) adjacency matrices. Here a new technique is developed for the general case, breaking the limitation of star complement technique for the standard \(0-1\) adjacency matrices.Between graphical zonotope and graph-associahedronhttps://zbmath.org/1517.050692023-09-22T14:21:46.120933Z"Pesovic, Marko"https://zbmath.org/authors/?q=ai:pesovic.marko"Stojadinovic, Tanja"https://zbmath.org/authors/?q=ai:stojadinovic.tanjaSummary: This manuscript introduces a finite collection of generalized permutohedra associated to a simple graph. The first polytope of this collection is the graphical zonotope of the graph, and the last is the graph-associahedron associated to it. We describe the weighted integer points enumerators for polytopes in this collection as Hopf algebra morphisms of combinatorial Hopf algebras of decorated graphs. In the last section, we study some properties related to \(\mathcal{H}\)-polytopes.A fast algorithm for source-wise round-trip spannershttps://zbmath.org/1517.050702023-09-22T14:21:46.120933Z"Zhu, Chun Jiang"https://zbmath.org/authors/?q=ai:zhu.chun-jiang"Han, Song"https://zbmath.org/authors/?q=ai:han.song"Lam, Kam-Yiu"https://zbmath.org/authors/?q=ai:lam.kam-yiuSummary: In this paper, we study the problem of fast constructions of source-wise round-trip spanners in weighted directed graphs. For a source vertex set \(S \subseteq V\) in a graph \(G(V, E)\), an \(S\)-sourcewise round-trip spanner of \(G\) of stretch \(k\) is a subgraph \(H\) of \(G\) such that for every pair of vertices \(u, v \in S \times V\), their round-trip distance in \(H\) is at most \(k\) times of their round-trip distance in \(G\). We show that for a graph \(G(V, E)\) with \(n\) vertices and \(m\) edges, an \(s\)-sized source vertex set \(S \subseteq V\) and an integer \(k > 1\), there exists an algorithm that in time \(O(m s^{1 / k} \log^5 n)\) constructs an \(S\)-sourcewise round-trip spanner of stretch \(O(k \log n)\) and \(O(n s^{1 / k} \log^2 n)\) edges with high probability. Compared to the fast algorithms for constructing all-pairs round-trip spanners [\textit{J. Pachocki} et al., in: Proceedings of the 29th annual ACM-SIAM symposium on discrete algorithms, SODA 2018, New Orleans, LA, USA, January 7--10, 2018. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM); New York, NY: Association for Computing Machinery (ACM). 1374--1392 (2018; Zbl 1403.68173); \textit{S. Chechik} et al., in: Proceedings of the 52nd annual ACM SIGACT symposium on theory of computing, STOC '20, Chicago, IL, USA, June 22--26, 2020. New York, NY: Association for Computing Machinery (ACM). 1010--1023 (2020; Zbl 07298306)], our algorithm improves the running time and the number of edges in the spanner when \(k\) is super-constant. Compared with the existing algorithm for constructing source-wise round-trip spanners [\textit{C. J. Zhu} and \textit{K.-Y. Lam}, Inf. Process. Lett. 124, 42--45 (2017; Zbl 1416.05272)], our algorithm significantly improves their construction time \(\Omega(\min \{m s, n^\omega \})\) (where \(\omega \in [2, 2.373)\) and 2.373 is the matrix multiplication exponent) to nearly linear \(O(m s^{1 / k} \log^5 n)\), at the expense of paying an extra \(O(\log n)\) in the stretch. As an important building block of the algorithm, we develop a graph partitioning algorithm to partition \(G\) into clusters of bounded radius and prove that for every \(u, v \in S \times V\) at small round-trip distance, the probability of separating them in different clusters is small. The algorithm takes the size of \(S\) as input and does not need the knowledge of \(S\). With the algorithm and a reachability vertex size estimation algorithm, we show that the recursive algorithm for constructing standard round-trip spanners [Pachocki et al., loc. cit.] can be adapted to the source-wise setting. We rigorously prove the correctness and computational complexity of the adapted algorithms. Finally, we show how to remove the dependence on the edge weight in the source-wise case.The automorphism group and fixing number of the orthogonality graph of the full matrix ringhttps://zbmath.org/1517.050712023-09-22T14:21:46.120933Z"Chen, Zhengxin"https://zbmath.org/authors/?q=ai:chen.zhengxin"Wang, Yu"https://zbmath.org/authors/?q=au:Wang, YuSummary: Let \(n \geq 3\), \(M_n(F)\) be the set of all \(n \times n\) matrices over a finite field \(F\), and \(R_n(F)\) the subset of \(M_n(F)\) consisting of all rank one matrices. In this paper, we first determine the automorphism group and the fixing number of the orthogonality graph of \(R_n(F)\), and then characterize the automorphism group and the fixing number of the orthogonality graph of \(M_n(F)\).Locally bi-2-transitive graphs and cycle-regular graphs, and the answer to a 2001 problem posed by Fouquet and Hahnhttps://zbmath.org/1517.050722023-09-22T14:21:46.120933Z"Conder, Marston"https://zbmath.org/authors/?q=ai:conder.marston-d-e"Zhou, Jin-Xin"https://zbmath.org/authors/?q=ai:zhou.jinxinSummary: A vertex-transitive but not edge-transitive graph \(\Gamma\) is called locally bi-2-transitive if the stabiliser \(S\) in the full automorphism group of \(\Gamma\) of every vertex \(v\) of \(\Gamma\) has two orbits of equal size on the neighbourhood of \(v\), and \(S\) acts 2-transitively on each of these two orbits. Also a graph is called cycle-regular if the number of cycles of a given length passing through a given edge in the graph is a constant, and a graph with girth \(g\) is called edge-girth-regular if the number of cycles of length \(g\) passing through any edge in the graph is a constant. In this paper, we prove that a graph of girth 3 is edge-girth-regular and locally bi-2-transitive if and only if \(\Gamma\) is the line graph of a semi-symmetric locally 3-transitive graph. Then as an application, we prove that every tetravalent edge-girth-regular locally bi-2-transitive graph of girth 3 is cycle-regular. This shows that vertex-transitive cycle-regular graphs need not to be edge-transitive, and hence resolves the problem posed by \textit{J.-L. Fouquet} and \textit{G. Hahn} [Discrete Appl. Math. 113, No. 2--3, 261--264 (2001; Zbl 0990.05086)] at the end of their paper.Binding number for coprime graph of groupshttps://zbmath.org/1517.050732023-09-22T14:21:46.120933Z"Mallika, A."https://zbmath.org/authors/?q=ai:mallika.a"Ahamed Thamimul Ansari, J."https://zbmath.org/authors/?q=ai:ahamed-thamimul-ansari.jSummary: Let \(G\) be a finite group with identity \(e\). The coprime of \(G\), \(\Gamma_G\) is a graph with \(G\) as the vertex set and two distinct vertices \(u\) and \(v\) are adjacent if and only if \((|u|, |v|)=1\). In this paper, we characterize the groups for which the binding number of coprime graph is 1 and investigate its bound range.Digital time: a finite field, \(T_\mathbb{F} \)https://zbmath.org/1517.050742023-09-22T14:21:46.120933Z"Manikandan, G."https://zbmath.org/authors/?q=ai:manikandan.g"Rao, K. Srinivasa"https://zbmath.org/authors/?q=ai:rao.k-srinivasa|rao.konda-srinivasaSummary: Digital time was defined [\textit{K. Srinivasa Rao} and \textit{P. Pundir}, South East Asian J. Math. Math. Sci. 13, No. 1, 1--9 (2017; Zbl 1378.20040)] with three two-digit positions as \(h_2h_1 : m_2m_1 : s_2s_1\). It was identified with appropriate restricted place values on the hours \((H)\), minutes \((M)\) and seconds \((S)\) shown to be \(86400\)-element cyclic Time Group, \(T_G\). Here it is shown to be a finite time field, \(T_\mathbb{F} \). A palindromic sequence of \(119\)-elements and its sub-sequences are shown to be consequences of \(T_F\).Prime ideal sum graph of a commutative ringhttps://zbmath.org/1517.050752023-09-22T14:21:46.120933Z"Saha, Manideepa"https://zbmath.org/authors/?q=ai:saha.manideepa"Das, Angsuman"https://zbmath.org/authors/?q=ai:das.angsuman"Çelikel, Ece Yetkin"https://zbmath.org/authors/?q=ai:celikel.ece-yetkin"Abdioğlu, Cihat"https://zbmath.org/authors/?q=ai:abdioglu.cihatThe prime ideal sum graph of a commutative ring with identity \(R\), denoted by PIS(\(R\)), is a graph whose vertices are nonzero proper ideals of \(R\) and two distinct vertices \(I\) and \(J\) are adjacent if and only if \(I + J\) is a prime ideal of \(R\). In this paper, the authors study some connections between the graph-theoretic properties of this graph and some algebraic properties of rings. The clique number, the girth, the chromatic number and the domination number of the prime ideal sum graph for some classes of rings are studied. For example, if \(R\) has \(k\) prime ideals, then PIS(\(R\)) is either acyclic or girth(PIS(\(R\))) \(\leq 2k\). It is observed that under which condition PIS(\(R\)) is complete: PIS(\(R\)) is complete if and only if \(R\) is a local ring and every proper non-prime ideal is a minimal ideal. Moreover, the diameter of PIS(\(R\)) was evaluated. For example, if \(R\) is an integral domain, then diam(PIS(\(R\))) \(\leq 4\).
Reviewer: Ioan Tomescu (Bucureşti)Commuting graph of \(CA\)-groupshttps://zbmath.org/1517.050762023-09-22T14:21:46.120933Z"Torktaz, Mehdi"https://zbmath.org/authors/?q=ai:torktaz.mehdi"Ashrafi, Ali Reza"https://zbmath.org/authors/?q=ai:ashrafi.ali-rezaSummary: A group \(G\) is called a \(CA\)-group, if all the element centralizers of \(G\) are abelian and the commuting graph of \(G\) with respect to a subset \(A\) of \(G\), denoted by \(\Gamma (G, A)\), is a simple undirected graph with vertex set \(A\) and two distinct vertices \(a\) and \(b\) are adjacent if and only if \(ab = ba\). The aim of this paper is to generalize results of a recently published paper of \textit{F. Ali} et al. [Commun. Algebra 44, No. 6, 2389--2401 (2016; Zbl 1339.05176)] to the case that \(G\) is an \(CA\)-group.On the parameterized complexity of counting small-sized minimum \((S,T)\)-cutshttps://zbmath.org/1517.050772023-09-22T14:21:46.120933Z"Bergé, Pierre"https://zbmath.org/authors/?q=ai:berge.pierre"Bouaziz, Wassim"https://zbmath.org/authors/?q=ai:bouaziz.wassim"Rimmel, Arpad"https://zbmath.org/authors/?q=ai:rimmel.arpad"Tomasik, Joanna"https://zbmath.org/authors/?q=ai:tomasik.joannaSummary: The counting of minimum edge \((S,T)\)-cuts in undirected graphs, parameterized by the size \(p\) of these cuts, is FPT. The best performance in the literature is \(O^\ast (2^{O(p^2)})\). We treat a more general problem of counting minimum \((S,T)\)-cuts composed of vertices instead of edges. We propose an FPT algorithm with running time \(O^\ast(2^{O(p\log p)})\). As it may be applied to the edge version as well, we improve the time complexity of the minimum edge \((S,T)\)-cuts counting.Counting tournament score sequenceshttps://zbmath.org/1517.050782023-09-22T14:21:46.120933Z"Claesson, Anders"https://zbmath.org/authors/?q=ai:claesson.anders"Dukes, Mark"https://zbmath.org/authors/?q=ai:dukes.mark"Franklín, Atli Fannar"https://zbmath.org/authors/?q=ai:franklin.atli-fannar"Stefánsson, Sigurður Örn"https://zbmath.org/authors/?q=ai:stefansson.sigurdur-ornSummary: The score sequence of a tournament is the sequence of the out-degrees of its vertices arranged in nondecreasing order. The problem of counting score sequences of a tournament with \(n\) vertices is more than 100 years old [\textit{P. A. MacMahon}, Quart. J. 49, 1--36 (1920; JFM 47.0090.02)]. In 2013 Hanna conjectured a surprising and elegant recursion for these numbers. We settle this conjecture in the affirmative by showing that it is a corollary to our main theorem, which is a factorization of the generating function for score sequences with a distinguished index. We also derive a closed formula and a quadratic time algorithm for counting score sequences.Graphs and unicyclic graphs with extremal number of connected induced subgraphshttps://zbmath.org/1517.050792023-09-22T14:21:46.120933Z"Dossou-Olory, Audace A. V."https://zbmath.org/authors/?q=ai:dossou-olory.audace-a-vSummary: Over all graphs (or unicyclic graphs) of a given order, we characterise those graphs that minimise or maximise the number of connected induced subgraphs. For each of these classes, we find that the graphs that minimise the number of connected induced subgraphs coincide with those that are known to maximise the Wiener index (sum of the distances between all unordered pairs of vertices), and vice versa. For every \(k\), we also determine the connected graphs that are extremal with respect to the number of \(k\)-vertex connected induced subgraphs. We show that, in contrast to the minimum which is uniquely realised by the path, the maximum value is attained by a rich class of connected graphs.Enumeration of subtrees and BC-subtrees with maximum degree no more than \(k\) in treeshttps://zbmath.org/1517.050802023-09-22T14:21:46.120933Z"Yang, Yu"https://zbmath.org/authors/?q=ai:yang.yu.3"Li, Xiao-xiao"https://zbmath.org/authors/?q=ai:li.xiaoxiao"Jin, Meng-yuan"https://zbmath.org/authors/?q=ai:jin.meng-yuan"Li, Long"https://zbmath.org/authors/?q=ai:li.long.1"Wang, Hua"https://zbmath.org/authors/?q=ai:wang.hua.2"Zhang, Xiao-Dong"https://zbmath.org/authors/?q=ai:zhang.xiaodongSummary: The subtrees and BC-subtrees (subtrees where any two leaves are at even distance apart) have been intensively studied in recent years. Such structures, under special constraints on degrees, have a wide range of applications in many fields. By way of an approach based on generating functions, we present novel recursive algorithms for enumerating various subtrees and BC-subtrees of maximum degree \(\leq k\) in trees. The algorithms are explained through detailed examples. We also briefly discuss, in trees, the densities of subtrees (resp. BC-subtrees) with maximum degree \(\leq k\) among all subtrees (resp. BC-subtrees). For a tree of order \(n\), the novelly proposed algorithms have multiple advantages. (1) Novel \((k + 2)\) (resp. \((2 k + 3))\) variable generating functions were introduced to construct the algorithms. (2) The proposed algorithms solved the fast enumerating problem of subtree (resp. BC-subtrees) with maximum degree constraint, and also make the subtree (resp. BC-subtrees) enumerating algorithms proposed by \textit{W. Yan} and \textit{Y.-N. Yeh} [Theor. Comput. Sci. 369, No. 1--3, 256--268 (2006; Zbl 1140.05308)] (resp. \textit{Y. Yang} et al. [ibid. 580, 59--74 (2015; Zbl 1311.05085)]) a special case of ours with \(k = n - 1\). (3) The time complexity of our algorithm for subtree (resp. BC-subtrees) is \(O(k n)\) (resp. \(O(k n^2))\), which is much faster than the \(O( n^2)\) (resp. \(O(k n^3))\) time method based on algorithm proposed in [Yan and Yeh, loc. cit.] (resp. [Yan et al., loc. cit.]).Chromatic Schultz and Gutman polynomials of Jahangir graphs \(J_{2, m}\) and \(J_{3, m}\)https://zbmath.org/1517.050812023-09-22T14:21:46.120933Z"Shaheen, Ramy"https://zbmath.org/authors/?q=ai:shaheen.ramy-s"Mahfud, Suhail"https://zbmath.org/authors/?q=ai:mahfud.suhail"Alhawat, Qays"https://zbmath.org/authors/?q=ai:alhawat.qaysSummary: Topological polynomial and indices based on the distance between the vertices of a connected graph are widely used in the chemistry to establish relation between the structure and the properties of molecules. In a similar way, chromatic versions of certain topological indices and the related polynomial have also been discussed in the recent literature. In this paper, we present the chromatic Schultz and Gutman polynomials and the expanded form of the Hosoya polynomial and chromatic Schultz and Gutman polynomials, and then we derive these polynomials for special cases of Jahangir graphs.Saturation of ordered graphshttps://zbmath.org/1517.050822023-09-22T14:21:46.120933Z"Bošković, Vladimir"https://zbmath.org/authors/?q=ai:boskovic.vladimir"Keszegh, Balázs"https://zbmath.org/authors/?q=ai:keszegh.balazsSummary: Recently, the saturation problem of 0-1 matrices has gained a lot of attention. This problem can be regarded as a saturation problem of ordered bipartite graphs. Motivated by this, we initiate the study of the saturation problem of ordered and cyclically ordered graphs. We prove that dichotomy also holds in these two cases, i.e., for a (cyclically) ordered graph its saturation function is either bounded or linear. We also determine the order of magnitude for large classes of (cyclically) ordered graphs, giving infinitely many examples exhibiting both possible behaviors, answering a problem of Pálvölgyi. In particular, in the ordered case we define a natural subclass of ordered matchings, the class of linked matchings, and we start their systematic study, concentrating on linked matchings with at most three links and prove that many of them have bounded saturation function. In both the ordered and cyclically ordered case we also consider the semisaturation problem, where dichotomy holds as well and we can even fully characterize the graphs that have bounded semisaturation function.Graphs of fixed order and size with maximal \(A_\alpha\)-indexhttps://zbmath.org/1517.050832023-09-22T14:21:46.120933Z"Chang, Ting-Chung"https://zbmath.org/authors/?q=ai:chang.ting-chung"Tam, Bit-Shun"https://zbmath.org/authors/?q=ai:tam.bit-shunSummary: For any real number \(\alpha \in [0,1]\), by the \(A_\alpha\)-matrix of a graph \(G\) we mean the matrix \(A_\alpha(G) = \alpha D(G) +(1 - \alpha) A(G)\), where \(A(G)\) and \(D(G)\) are the adjacency matrix and the diagonal matrix of vertex degrees of \(G\), respectively. The largest eigenvalue of \(A_\alpha(G)\) is called the \(A_\alpha \)-index of \(G\). In this paper, we settle the problem of characterizing graphs which attain the maximum \(A_\alpha \)-index over \(\mathcal{G}(n, n + k)\), the class of graphs with \(n\) vertices and \(n + k\) edges, for \(- 1 \leq k \leq n - 3\) and \(\frac{ 1}{ 2} \leq \alpha < 1\). The following result is obtained: for \(- 1 \leq k \leq n - 3\), when \(\frac{1}{2} \leq \alpha < 1\), \(H_{n,k}\) is the unique graph in \(\mathcal{G}(n, n + k)\) that maximizes the \(A_\alpha \)-index, except when \((n, k) = (4, - 1)\), \((n, 2)\) or \((7, 3)\) and \(\alpha = \frac{ 1}{ 2} \), or \((n, k) = (5, 1)\) and \(\alpha \in [\frac{ 1}{ 2}, \frac{ 35 - \sqrt{ 409}}{ 24}]\). When \((n, k, \alpha) = (4, - 1, \frac{ 1}{ 2})\), the optimal graphs are \(H_{4,-1}\) and \(K_3 \cup K_1\); when \((n, k, \alpha) = (n, 2, \frac{1}{2})\), the optimal graphs are \(H_{n , 2}\) and \(G_{n , 2} \); when \((n, k, \alpha) = (5, 1, \frac{ 35 - \sqrt{ 409}}{ 24})\), the optimal graphs are \(H_{5,1}\) and \(K_4 \cup K_1\); when \((n, k, \alpha) = (7, 3, \frac{ 1}{ 2})\), the optimal graphs are \(H_{7 , 3}\) and \(K_5 \cup 2 K_1\); when \((n, k) = (5, 1)\) and \(\frac{ 1}{ 2} \leq \alpha < \frac{ 35 - \sqrt{ 409}}{ 24}\), \(K_4 \cup K_1\) is the unique graph that maximizes the \(A_\alpha\)-index. Our work completes the corresponding work of the authors [ibid. 432, No. 7, 1708--1733 (2010; Zbl 1231.05166)] and \textit{M. Zhai} et al. [Discrete Math. 345, No. 1, Article ID 112669, 13 p. (2022; Zbl 1480.05088)] for the special case \(\alpha = \frac{1}{2} \). As a by-product, we provide a new proof for the known result that for any positive integer \(m\) and any real number \(\alpha \in [\frac{1}{2}, 1)\), if \((m, \alpha) \neq(3, \frac{1}{2})\), then a graph maximizes the \(A_\alpha \)-index over all graphs with \(m\) edges if and only if it is the union of \(K_{1 , m}\) with a (possibly empty) null graph; a graph maximizes the \(A_{\frac{ 1}{ 2}} \)-index over all graphs with three edges if and only if it is the union of \(K_{1 , 3}\) or \(K_3\) with a (possibly empty) null graph. Some open questions are also posed.Maximal knotless graphshttps://zbmath.org/1517.050842023-09-22T14:21:46.120933Z"Eakins, Lindsay"https://zbmath.org/authors/?q=ai:eakins.lindsay"Fleming, Thomas"https://zbmath.org/authors/?q=ai:fleming.thomas-r"Mattman, Thomas"https://zbmath.org/authors/?q=ai:mattman.thomas-wSummary: A graph is maximal knotless if it is edge maximal for the property of knotless embedding in \(\mathbb{R}^3\). We show that such a graph has at least \(\frac74 |V|\) edges, and construct an infinite family of maximal knotless graphs with \(|E| < \frac52|V|\). With the exception of \(|E| = 22\), we show that for any \(|E| \geq 20\) there exists a maximal knotless graph of size \(|E|\). We classify the maximal knotless graphs through nine vertices and 20 edges. We determine which of these maxnik graphs are the clique sum of smaller graphs and construct an infinite family of maxnik graphs that are not clique sums.On the Turán number of generalized theta graphshttps://zbmath.org/1517.050852023-09-22T14:21:46.120933Z"Liu, Xiao-Chuan"https://zbmath.org/authors/?q=ai:liu.xiaochuan"Yang, Xu"https://zbmath.org/authors/?q=ai:yang.xuLet \(\Theta_{ k_1,\dots,k_\ell}\) denote the generalized theta graph, which consists of \(\ell\) internally disjoint paths with lengths \(k_1, \dots , k_\ell\) , connecting two fixed vertices. In this paper, the authors estimate the corresponding extremal number ex(\(n, \Theta_{ k_1,\dots ,k_\ell}\) ). When the lengths of all paths have the same parity and at most one path has length 1, ex(\(n, \Theta_{ k_1,\dots ,k_\ell}\) ) is \(O(n^{ 1+1/k^\ast}\) ), where \(2k^\ast\) is the length of the smallest cycle in \(\Theta_{ k_1, \dots,k_\ell}\). They also establish a matching lower bound in the particular case of ex(\(n, \Theta_{ 3,5,5}\)) which is equal to \(\Theta (n^{5/4})\).
Reviewer: Ioan Tomescu (Bucureşti)Typical large graphs with given edge and triangle densitieshttps://zbmath.org/1517.050862023-09-22T14:21:46.120933Z"Neeman, Joe"https://zbmath.org/authors/?q=ai:neeman.joe"Radin, Charles"https://zbmath.org/authors/?q=ai:radin.charles"Sadun, Lorenzo"https://zbmath.org/authors/?q=ai:sadun.lorenzo-aSummary: The analysis of large simple graphs with extreme values of the densities of edges and triangles has been extended to the statistical structure of typical graphs of fixed intermediate densities, by the use of large deviations of Erdős-Rényi graphs. We prove that the typical graph exhibits sharp singularities as the constraining densities vary between different curves of extreme values, and we determine the precise nature of the singularities. The extension to graphs with fixed densities of edges and \(k\)-cycles for odd \(k>3\) is straightforward and we note the simple changes in the proof.On the running time of hypergraph bootstrap percolationhttps://zbmath.org/1517.050872023-09-22T14:21:46.120933Z"Noel, Jonathan A."https://zbmath.org/authors/?q=ai:noel.jonathan-a"Ranganathan, Arjun"https://zbmath.org/authors/?q=ai:ranganathan.arjunSummary: Given \(r\geqslant 2\) and an \(r\)-uniform hypergraph \(F\), the \(F\)-bootstrap process starts with an \(r\)-uniform hypergraph \(H\) and, in each time step, every hyperedge which ``completes'' a copy of \(F\) is added to \(H\). The maximum running time of this process has been recently studied in the case that \(r=2\) and \(F\) is a complete graph by \textit{B. Bollobás} et al. [Electron. J. Comb. 24, No. 2, Research Paper P2.16, 20 p. (2017; Zbl 1361.05135)], \textit{K. Matzke} [``The saturation time of graph bootstrap percolation'', Preprint, \url{arXiv:1510.06156v2}] and \textit{J. Balogh} et al. [``The maximum length of \(K_r\)-bootstrap percolation'', Preprint, \url{arXiv:1907.04559v1}]. We consider the case that \(r\geqslant 3\) and \(F\) is the complete \(r\)-uniform hypergraph on \(k\) vertices. Our main results are that the maximum running time is \(\Theta (n^r)\) if \(k\geqslant r+2\) and \(\Omega(n^{r-1})\) if \(k=r+1\). For the case \(k=r+1\), we conjecture that our lower bound is optimal up to a constant factor when \(r=3\), but suspect that it can be improved by more than a constant factor for large \(r\).On unicyclic non-bipartite graphs with tricyclic inverseshttps://zbmath.org/1517.050882023-09-22T14:21:46.120933Z"Kalita, Debajit"https://zbmath.org/authors/?q=ai:kalita.debajit"Sarma, Kuldeep"https://zbmath.org/authors/?q=ai:sarma.kuldeepSummary: The class of unicyclic non-bipartite graphs with unique perfect matching, denoted by \(\mathcal{U}\), is considered in this article. This article provides a complete characterization of unicyclic graphs in \(\mathcal{U}\) which possess tricyclic inverses.Successive vertex orderings of fully regular graphshttps://zbmath.org/1517.050892023-09-22T14:21:46.120933Z"Fang, Lixing"https://zbmath.org/authors/?q=ai:fang.lixing"Huang, Hao"https://zbmath.org/authors/?q=ai:huang.hao.4|huang.hao|huang.hao.1|huang.hao.2|huang.hao.3"Pach, János"https://zbmath.org/authors/?q=ai:pach.janos"Tardos, Gábor"https://zbmath.org/authors/?q=ai:tardos.gabor"Zuo, Junchi"https://zbmath.org/authors/?q=ai:zuo.junchiSummary: A graph \(G=(V,E)\) is called fully regular if for every independent set \(I\subset V\), the number of vertices in \(V\setminus I\) that are not connected to any element of \(I\) depends only on the size of \(I\). A linear ordering of the vertices of \(G\) is called successive if for every \(i\), the first \(i\) vertices induce a connected subgraph of \(G\). We give an explicit formula for the number of successive vertex orderings of a fully regular graph.
As an application of our results, we give alternative proofs of two theorems of \textit{R. Stanley} [``Counting `connected' edge orderings (shellings) of the complete graph'', \url{https://mathoverflow.net/questions/297411/counting-connected-edge-orderings-shellings-of-the-complete-graph}] and \textit{Y. Gao} and \textit{J. Peng} [J. Algebr. Comb. 54, No. 1, 17--37 (2021; Zbl 07380961)], determining the number of linear edge orderings of complete graphs and complete bipartite graphs, respectively, with the property that the first \(i\) edges induce a connected subgraph.
As another application, we give a simple product formula for the number of linear orderings of the hyperedges of a complete 3-partite 3-uniform hypergraph such that, for every \(i\), the first \(i\) hyperedges induce a connected subgraph. We found similar formulas for complete (non-partite) 3-uniform hypergraphs and in another closely related case, but we managed to verify them only when the number of vertices is small.The complexity of 2-vertex-connected orientation in mixed graphshttps://zbmath.org/1517.050902023-09-22T14:21:46.120933Z"Hörsch, Florian"https://zbmath.org/authors/?q=ai:horsch.florian"Szigeti, Zoltán"https://zbmath.org/authors/?q=ai:szigeti.zoltanSummary: We consider two possible extensions of a theorem of \textit{C. Thomassen} [in: Combinatorial mathematics: Proceedings of the third international conference, held in New York, USA, June 10--14, 1985. New York: New York Academy of Sciences. 402--412 (1989; Zbl 0709.05030)] characterizing the graphs admitting a 2-vertex-connected orientation. First, we show that the problem of deciding whether a mixed graph has a 2-vertex-connected orientation is NP-hard. This answers a question of \textit{J. Bang-Jensen} et al. [J. Graph Theory 87, No. 3, 285--304 (2018; Zbl 1386.05121)]. For the second part, we call a directed graph \(D=(V,A)2T\)-connected for some \(T\subseteq V\) if \(D\) is 2-arc-connected and \(D-v\) is strongly connected for all \(v\in T\). We deduce a characterization of the graphs admitting a \(2T\)-connected orientation from the theorem of Thomassen [loc. cit.].The generalized 3-connectivity of burnt pancake graphs and godan graphshttps://zbmath.org/1517.050912023-09-22T14:21:46.120933Z"Wang, Jing"https://zbmath.org/authors/?q=ai:wang.jing.3"Zhang, Zuozheng"https://zbmath.org/authors/?q=ai:zhang.zuozheng"Huang, Yuanqiu"https://zbmath.org/authors/?q=ai:huang.yuanqiuSummary: The generalized \(k\)-connectivity of a graph G, denoted by \(\kappa_k(G)\) is the minimum number of internally edge disjoint S-trees for any \(S \subseteq V(G)\) and \(|S| =k\). The generalized \(k\)-connectivity is a natural extension of the classical connectivity and plays a key role in applications related to the modern interconnection networks. The burnt pancake graph \(\mathrm{BP}_n\) and the godan graph \(\mathrm{EA}_n\) are two kinds of Cayley graphs which posses many desirable properties. In this paper, we investigate the generalized 3-connectivity of \(\mathrm{BP}_n\) and \(\mathrm{EA}_n\). We show that \(\kappa_3 (\mathrm{BP}_n)=n-1\) where \(n \geq 2\) and \(\kappa_3(\mathrm{EA}_n)= n-1\) where \(n \geq 3\).The spectrum of triangle-free graphshttps://zbmath.org/1517.050922023-09-22T14:21:46.120933Z"Balogh, József"https://zbmath.org/authors/?q=ai:balogh.jozsef"Clemen, Felix Christian"https://zbmath.org/authors/?q=ai:clemen.felix-christian"Lidický, Bernard"https://zbmath.org/authors/?q=ai:lidicky.bernard"Norin, Sergey"https://zbmath.org/authors/?q=ai:norine.serguei"Volec, Jan"https://zbmath.org/authors/?q=ai:volec.janSummary: Denote by \(q_n(G)\) the smallest eigenvalue of the signless Laplacian matrix of an \(n\)-vertex graph \(G\). \textit{S. Brandt} [Discrete Math. 183, No. 1--3, 17--25 (1998; Zbl 0895.05042)] conjectured that for regular triangle-free graphs \(q_n(G)\leq\frac{4n}{25}\). We prove a stronger result: If \(G\) is a triangle-free graph, then \(q_n(G)\leq\frac{15n}{94}<\frac{4n}{25}\). Brandt's conjecture is a subproblem of two famous conjectures of \textit{P. Erdős} [ibid. 165--166, 227--231 (1997; Zbl 0872.05020); in: Graph theory and related topics. Proceedings of the conference held in honour of Professor W. T. Tutte on the occasion of his sixtieth birthday, University of Waterloo, July 5--9, 1977. New York - San Francisco - London: Academic Press, A Subsidiary of Harcourt Brace Jovanovich, Publishers. 153--163 (1979; Zbl 0457.05024)]: (1) Sparse-half-conjecture: Every \(n\)-vertex triangle-free graph has a subset of vertices of size \(\lceil\frac{n}{2}\rceil\) spanning at most \(n^2/50\) edges. (2) Every \(n\)-vertex triangle-free graph can be made bipartite by removing at most \(n^2/25\) edges. In our proof we use linear algebraic methods to upper bound \(q_n(G)\) by the ratio between the number of induced paths with 3 and 4 vertices. We give an upper bound on this ratio via the method of flag algebras.Eigenvalue sum estimates for lattice subgraphshttps://zbmath.org/1517.050932023-09-22T14:21:46.120933Z"Bauer, Frank"https://zbmath.org/authors/?q=ai:bauer.frank"Lippner, Gabor"https://zbmath.org/authors/?q=ai:lippner.gaborThis paper presents discrete analogues of the previously obtained estimates for the average of the first \(k\) Dirichlet eigenvalues by \textit{P. Li} and \textit{S.-T. Yau} [Commun. Math. Phys. 88, 309--318 (1983; Zbl 0554.35029)], that is,
\[
\frac{1}{k}\sum_{i=1}^{k}\lambda_i(\Omega)\geq \frac{n}{n+2} C_n \left(\frac{k}{V(\Omega)}\right)^{\frac{2}{n}}
\]
where \(V(\Omega)\) is the volume of any bounded open subset \(\Omega\subseteq \mathbb{R}^n\) and \(C_n=(2\pi)^2V_n^{-\frac{2}{n}}\) is the Weyl constant with \(V_n\) being the volume of the unit ball in \(\mathbb{R}^n\).
The authors consider \(\Omega\) to be a finite induced subgraph in the infinite \(d\)-dimensional integer lattice \(\mathbb{Z}^d\), viewed as a graph.
Consider the Dirichlet eigenvalue problem \(\Delta_\Omega^{\mathcal{D}}\phi=-\lambda\phi\),
where \(\Delta_\Omega^{\mathcal{D}}\) is the Laplacian with Dirichlet boundary conditions. There are \(|\Omega|\) eigenvalues (with multiplicities) of the Dirichlet problem, which are all real and positive. Denote the eigenvalues by \(0<\lambda_1\leq \lambda_2\leq\dots\leq \lambda_{|\Omega|}\). Thus, for \(1\leq k\leq |\Omega|\min(1, V_d)\),
\[
\frac{1}{k}\sum_{j=1}^{k}\lambda_j\leq\frac{4\pi^2d}{d+2}\left(\frac{k}{V_d}|\Omega|\right)^{\frac{2}{d}}+\frac{\partial\Omega}{|\Omega|}
\]
while for \(1\leq k\leq|\Omega|\min\left(1,V_d\left(\sqrt{6}/(2\pi)\right)^d\right)\)
\[\frac{4\pi^2d}{d+2}\left(\frac{k}{V_d|\Omega|}\right)^{\frac{2}{d}}-\frac{\pi^4d}{3(d+4)}\left(\frac{k}{|\Omega|V_d}\right)^{\frac{4}{d}}\leq \frac{1}{k}\sum_{j=1}^k\lambda_j.
\]
The proof of the upper bound on the sum of the first \(k\) eigenvalues of the Dirichlet-Laplace operator on \(\Omega\) (Theorem 3.3) is established by first presenting a general lemma about eigenspaces (Lemma 3.1). Let \(L\) be a self-adjoint, positive semidefinite operator on a finite-dimensional, Hermitian, complex vector space \(W\) with Hermitian inner product \(\langle , \rangle\). Let \(0\leq\gamma_1\leq\dots\leq \gamma_1\) denote its eigenvalues, and choose an orthonormal basis of eigenfunctions \(f_i:i=1,2,\dots, s\), where \(f_i\) corresponds to \(\gamma_i\). Then for any \(1\leq k\leq s\) and any vector \(g\in W\) one has
\[
\gamma_{k+1}\langle g,g\rangle \leq \langle g, Lg\rangle +\sum_{j=1}^{k}\left(\gamma_{k+1}-\gamma_j\right)\left|\langle g,f_j\rangle \right|^2.
\]
Then this lemma is used by ``averaging'' it over a set of carefully chosen \(g\)'s (Lemma 3.2). To get the lower bound (Theorem 3.6), the authors use an adaptation of Li and Yau's method that involves expressing the eigenvalue sum as an integral and then using bounds on the integrand to get an estimate for the sum. The integral in the discrete case is slightly different from the one in the original continuous version, so a modified version of the lemma (Lemma 3.5) is proved to enable them to derive such lower bounds.
Reviewer: Roxanne Anunciado (Agusan del Norte)From independent sets and vertex colorings to isotropic spaces and isotropic decompositions: another bridge between graphs and alternating matrix spaceshttps://zbmath.org/1517.050942023-09-22T14:21:46.120933Z"Bei, Xiaohui"https://zbmath.org/authors/?q=ai:bei.xiaohui"Chen, Shiteng"https://zbmath.org/authors/?q=ai:chen.shiteng"Guan, Ji"https://zbmath.org/authors/?q=ai:guan.ji"Qiao, Youming"https://zbmath.org/authors/?q=ai:qiao.youming"Sun, Xiaoming"https://zbmath.org/authors/?q=ai:sun.xiaomingSummary: \textit{L. Lovász} [``On determinants, matchings, and random algorithms'', in: Proceedings of the conference on algebraic, arithmetic, and categorial methods in computation theory. Berlin: Akademie-Verlag. 565--574 (1979; Zbl 0446.68036)] built a bridge between graphs and alternating matrix spaces, in the context of perfect matchings. A similar connection between bipartite graphs and matrix spaces plays a key role in the recent resolutions of the noncommutative rank problem [\textit{A. Garg} et al., in: Proceedings of the 49th annual ACM SIGACT symposium on theory of computing, STOC 2017. New York, NY: Association for Computing Machinery (ACM). 397--409 (2017; Zbl 1372.65191); \textit{G. Ivanyos} et al., Comput. Complexity 26, No. 3, 717--763 (2017; Zbl 1421.13002)]. In this paper, we lay the foundation for another bridge between graphs and alternating matrix spaces, in the context of independent sets and vertex colorings. The corresponding structures in alternating matrix spaces are isotropic spaces and isotropic decompositions, both useful structures in group theory and manifold theory. We first show that the maximum independent set problem and the vertex \(c\)-coloring problem reduce to the maximum isotropic space problem and the isotropic \(c\)-decomposition problem, respectively. Next, we show that several topics and results about independent sets and vertex colorings have natural correspondences for isotropic spaces and decompositions. These include algorithmic problems, such as the maximum independent set problem for bipartite graphs, and exact exponential-time algorithms for the chromatic number, as well as mathematical questions, such as the number of maximal independent sets, and the relation between the maximum degree and the chromatic number. These connections lead to new interactions between graph theory and algebra. Some results have concrete applications to group theory and manifold theory, and we initiate a variant of these structures in the context of quantum information theory. Finally, we propose several open questions for further exploration.From independent sets and vertex colorings to isotropic spaces and isotropic decompositions: another bridge between graphs and alternating matrix spaceshttps://zbmath.org/1517.050952023-09-22T14:21:46.120933Z"Bei, Xiaohui"https://zbmath.org/authors/?q=ai:bei.xiaohui"Chen, Shiteng"https://zbmath.org/authors/?q=ai:chen.shiteng"Guan, Ji"https://zbmath.org/authors/?q=ai:guan.ji"Qiao, Youming"https://zbmath.org/authors/?q=ai:qiao.youming"Sun, Xiaoming"https://zbmath.org/authors/?q=ai:sun.xiaomingSummary: \textit{L. Lovász} [in: Fundamentals of computation theory -- FCT '79. Proceedings of the conference on algebraic, arithmetic, and categorial methods in computation theory held in Berlin/Wendisch-Rietz (GDR) September 17--21, 1979. Berlin: Akademie-Verlag. 565--574 (1979; Zbl 0446.68036)] built a bridge between graphs and alternating matrix spaces, in the context of perfect matchings. A similar connection between bipartite graphs and matrix spaces plays a key role in the recent resolutions of the non-commutative rank problem [\textit{A. Garg} et al., in: Proceedings of the 57th annual IEEE symposium on foundations of computer science, FOCS 2016, New Brunswick, NJ, USA, October 9--10, 2016. Los Alamitos, CA: IEEE Computer Society. 109--117 (2016; \url{doi:10.1109/FOCS.2016.95}); \textit{G. Ivanyos} et al., Comput. Complexity 26, No. 3, 717--763 (2017; Zbl 1421.13002)]. In this paper, we lay the foundation for another bridge between graphs and alternating matrix spaces, in the context of independent sets and vertex colorings. The corresponding structures in alternating matrix spaces are isotropic spaces and isotropic decompositions, both useful structures in group theory and manifold theory.
We first show that the maximum independent set problem and the vertex \(c\)-coloring problem reduce to the maximum isotropic space problem and the isotropic \(c\)-decomposition problem, respectively. Next, we show that several topics and results about independent sets and vertex colorings have natural correspondences for isotropic spaces and decompositions. These include algorithmic problems, such as the maximum independent set problem for bipartite graphs, and exact exponential-time algorithms for the chromatic number, as well as mathematical questions, such as the number of maximal independent sets, and the relation between the maximum degree and the chromatic number. These connections lead to new interactions between graph theory and algebra. Some results have concrete applications to group theory and manifold theory, and we initiate a variant of these structures in the context of quantum information theory. Finally, we propose several open questions for further exploration.
Dedicated to the memory of Ker-I Ko.
For the entire collection see [Zbl 1434.68035].Nonbacktracking spectral clustering of nonuniform hypergraphshttps://zbmath.org/1517.050962023-09-22T14:21:46.120933Z"Chodrow, Philip"https://zbmath.org/authors/?q=ai:chodrow.philip-s"Eikmeier, Nicole"https://zbmath.org/authors/?q=ai:eikmeier.nicole"Haddock, Jamie"https://zbmath.org/authors/?q=ai:haddock.jamieSummary: Spectral methods offer a tractable, global framework for clustering in graphs via eigenvector computations on graph matrices. Hypergraph data, in which entities interact on edges of arbitrary size, poses challenges for matrix representations and therefore for spectral clustering. We study spectral clustering for nonuniform hypergraphs based on the hypergraph nonbacktracking operator. After reviewing the definition of this operator and its basic properties, we prove a theorem of Ihara-Bass type which allows eigenpair computations to take place on a smaller matrix, often enabling faster computation. We then propose an alternating algorithm for inference in a hypergraph stochastic blockmodel via linearized belief-propagation which involves a spectral clustering step again using nonbacktracking operators. We provide proofs related to this algorithm that both formalize and extend several previous results. We pose several conjectures about the limits of spectral methods and detectability in hypergraph stochastic blockmodels in general, supporting these with in-expectation analysis of the eigenpairs of our operators. We perform experiments in real and synthetic data that demonstrate the benefits of hypergraph methods over graph-based ones when interactions of different sizes carry different information about cluster structure.Comparing the principal eigenvector of a hypergraph and its shadowshttps://zbmath.org/1517.050972023-09-22T14:21:46.120933Z"Clark, Gregory J."https://zbmath.org/authors/?q=ai:clark.gregory-j"Thomaz, Felipe"https://zbmath.org/authors/?q=ai:thomaz.felipe"Stephen, Andrew T."https://zbmath.org/authors/?q=ai:stephen.andrew-tSummary: Graphs (i.e., networks) have become an integral tool for the representation and analysis of relational data. Advances in data gathering have led to multi-relational data sets which exhibit greater depth and scope. In certain cases, this data can be modeled using a hypergraph. However, in practice analysts typically reduce the dimensionality of the data (whether consciously or otherwise) to accommodate a traditional graph model. In recent years spectral hypergraph theory has emerged to study the eigenpairs of the adjacency hypermatrix of a uniform hypergraph. We show how analyzing multi-relational data, via a hypermatrix associated to the aforementioned hypergraph, can lead to conclusions different from those when the data is projected down to its co-occurrence matrix. To this end we consider how the principal eigenvector of a hypergraph and its shadow can vary in terms of their spectral rankings, Pearson/Spearman correlation coefficient, and Chebyshev distance. In particular, we provide an example of a uniform hypergraph where the most central vertex (à la eigencentrality) changes depending on the order of the associated matrix. To the best of our knowledge this is the first known hypergraph to exhibit this property. We further show that the aforementioned eigenvectors have a high Pearson correlation but are uncorrelated under the Spearman correlation coefficient.A common variable minimax theorem for graphshttps://zbmath.org/1517.050982023-09-22T14:21:46.120933Z"Coifman, Ronald R."https://zbmath.org/authors/?q=ai:coifman.ronald-raphael"Marshall, Nicholas F."https://zbmath.org/authors/?q=ai:marshall.nicholas-f"Steinerberger, Stefan"https://zbmath.org/authors/?q=ai:steinerberger.stefanLet \(\mathcal{G}=\{G_1=(V,E_1),\ldots, G_m=(V,E_m)\}\) be a collection of \(m\) graphs defined on a common set of vertices \(V\) but with different edge sets \(E_1,\ldots, E_m\). Informally, a function \(f: V\rightarrow \mathbb{R}\) is smooth with respect to \(G_k=(V,E_k)\) if it varies little over adjacent vertices. The authors study the problem of understanding whether there exists a nonconstant function that is smooth with respect to all graphs in \(\mathcal{G}\), simultaneously, and how to find it if it exists.
Reviewer: Fenglei Tian (Rizhao)Distance matrix of a multi-block graph: determinant and inversehttps://zbmath.org/1517.050992023-09-22T14:21:46.120933Z"Das, Joyentanuj"https://zbmath.org/authors/?q=ai:das.joyentanuj"Mohanty, Sumit"https://zbmath.org/authors/?q=ai:mohanty.sumitA connected graph is called a multi-block graph if each of its blocks is a complete multipartite graph. In this paper, the authors consider the distance matrix of multi-block graphs with blocks whose distance matrices have a nonzero cofactor. In this case, if the distance matrix of a multi-block graph is invertible, they find the inverse as a rank one perturbation of a multiple of a Laplacian-like matrix. They also show that the inverse of the distance matrix for a class of multi-block graphs in which the distance matrix of one of its blocks has zero cofactor.
Reviewer: Fenglei Tian (Rizhao)On the multiplicities of digraph eigenvalueshttps://zbmath.org/1517.051002023-09-22T14:21:46.120933Z"Gavrilyuk, Alexander L."https://zbmath.org/authors/?q=ai:gavrilyuk.alexander-l"Suda, Sho"https://zbmath.org/authors/?q=ai:suda.shoSummary: We show various upper bounds for the order of a digraph (or a mixed graph) whose Hermitian adjacency matrix has an eigenspace of prescribed codimension. In particular, this generalizes the so-called absolute bound for (simple) graphs first shown by \textit{P. Delsarte} et al. [Geom. Dedicata 6, 363--388 (1977; Zbl 0376.05015)] and extended by \textit{F. K. Bell} and \textit{P. Rowlinson} [Bull. Lond. Math. Soc. 35, No. 3, 401--408 (2003; Zbl 1023.05097)]. In doing so, we also adapt the \textit{A. Blokhuis}' theory [Few-distance sets. Eindhoven: Technische Hogeschool Eindhoven (1983; Zbl 0516.05017)] of harmonic analysis in real hyperbolic spaces to that in complex hyperbolic spaces.Computing some Laplacian coefficients of forestshttps://zbmath.org/1517.051012023-09-22T14:21:46.120933Z"Ghalavand, Ali"https://zbmath.org/authors/?q=ai:ghalavand.ali"Ashrafi, Ali Reza"https://zbmath.org/authors/?q=ai:ashrafi.ali-rezaSummary: Let \(G\) be a finite simple graph with Laplacian polynomial \(\psi(G, \lambda) = \sum_{k = 0}^n(-1)^{n - k} c_k \lambda^k\). In an earlier paper, the coefficients \(c_{n - 4}\) and \(c_{n - 5}\) for forests with respect to some degree-based graph invariants were computed. The aim of this paper is to continue this work by giving an exact formula for the coefficient \(c_{n - 6}\).A study on determination of some graphs by Laplacian and signless Laplacian permanental polynomialshttps://zbmath.org/1517.051022023-09-22T14:21:46.120933Z"Khan, Aqib"https://zbmath.org/authors/?q=ai:khan.aqib"Panigrahi, Pratima"https://zbmath.org/authors/?q=ai:panigrahi.pratima"Panda, Swarup Kumar"https://zbmath.org/authors/?q=ai:panda.swarup-kumarSummary: The \textit{permanent} of an \(n \times n\) matrix \(M=(m_{ij})\) is defined as \(\operatorname{per}(M) = \sum_\sigma \prod^n_{i=1} m_{i \sigma (i)}\), where the sum is taken over all permutations \(\sigma\) of \(\{1,2, \dots, n\}\) The \textit{permanental polynomial} of \(M\), denoted by \(\psi (M;x)\) is \(\operatorname{per}(xl_n - M)\) where \(I_n\) is the identity matrix of order \(n\). Let \(G\) be a simple undirected graph on \(n\) vertices and its Laplacian and signless Laplacian matrices be \(L(G)\) and \(Q(G)\) respectively. The permanental polynomials \(\psi (L(G);x)\) and \(\psi (Q(G);x)\) are called the \textit{Laplacian permanental polynomial} and \textit{signless Laplacian permanental polynomial} of \(G\) respectively. A graph \(G\) is said to be \textit{determined by its (signless) Laplacian permanental polynomial} if all the graphs having the same (signless) Laplacian permanental polynomial with \(G\) are isomorphic to \(G\). A graph \(G\) is said to be \textit{combinedly determined by its Laplacian and signless Laplacian permanental polynomials} if all the graphs having \textit{(i)} the same Laplacian permanental polynomial as \(\psi (L(G);x)\) and \textit{(ii)} the same signless Laplacian permanental polynomial as \(\psi(Q(G);x)\), are isomorphic to \(G\). In this article we investigate the determination of some graphs, namely, star, wheel, friendship graphs and a particular kind of caterpillar graph \(S_n^{(r)}\) (whose all \(r\) non-pendant vertices have the same degree \(n)\) by their Laplacian and signless Laplacian permanental polynomials. We show that a kind of caterpillar graphs \(S_n^{(r)}\) (for \(r=2,3,4,5)\), wheel graph (up to 7 vertices) and friendship graph (up to 7 vertices) are determined by their (signless) Laplacian permanental polynomials. Further we prove that all friendship graphs and wheel graphs are combinedly determined by their Laplacian and signless Laplacian permanental polynomials.Combinatorial necessary conditions for regular graphs to induce periodic quantum walkshttps://zbmath.org/1517.051032023-09-22T14:21:46.120933Z"Kubota, Sho"https://zbmath.org/authors/?q=ai:kubota.shoSummary: We derive combinatorial necessary conditions for discrete-time quantum walks defined by regular mixed graphs to be periodic. One useful necessary condition is that if a \(k\)-regular mixed graph with \(n\) vertices is periodic, then \(2n/k\) must be an integer. As an application of this work, we determine periodicity of mixed complete graphs and mixed graphs with a prime number of vertices. Furthermore, we study periodicity of mixed strongly regular graphs and several classes of mixed distance-regular graphs, and extend existing results to mixed graphs.The weight distribution of hulls of binary codes from incidence matrices of complete graphshttps://zbmath.org/1517.051042023-09-22T14:21:46.120933Z"Kumwenda, Khumbo"https://zbmath.org/authors/?q=ai:kumwenda.khumbo"Namondwe, Caleb"https://zbmath.org/authors/?q=ai:namondwe.calebSummary: We characterise the weights of all codewords of the hull of the binary code spanned by rows of an incidence matrix \(B_n\) of the complete graph \(K_n\) for \(n \geq 4\) and \(n\) even. Among others, we show that every codeword of the hull is the sum of an even number of rows of \(B_n\) and that its weight depends on the number of rows required to span it. We also determine the weight enumerator and the MacWilliams identity for the hull.The minimum spectral radius of graphs with a given domination numberhttps://zbmath.org/1517.051052023-09-22T14:21:46.120933Z"Liu, Chang"https://zbmath.org/authors/?q=ai:liu.chang.1"Li, Jianping"https://zbmath.org/authors/?q=ai:li.jianping.2"Xie, Zheng"https://zbmath.org/authors/?q=ai:xie.zhengSummary: Let \(\mathbb{G}_{n,\gamma}\) be the set of simple and connected graphs on \(n\) vertices and with domination number \(\gamma \). The graph with minimum spectral radius among \(\mathbb{G}_{n,\gamma}\) is called the minimizer graph. In this paper, we first prove that the minimizer graph of \(\mathbb{G}_{n,\gamma}\) must be a tree. Moreover, for \(\gamma \in \{1, 2, 3, \lceil \frac{n}{3} \rceil, \lfloor \frac{n}{2} \rfloor \} \), we characterize all minimizer graphs in \(\mathbb{G}_{n,\gamma} \).Maxima of the \(Q\)-index of non-bipartite \(C_3\)-free graphshttps://zbmath.org/1517.051062023-09-22T14:21:46.120933Z"Liu, Ruifang"https://zbmath.org/authors/?q=ai:liu.ruifang"Miao, Lu"https://zbmath.org/authors/?q=ai:miao.lu"Xue, Jie"https://zbmath.org/authors/?q=ai:xue.jieSummary: A classic result in extremal graph theory, known as Mantel's theorem, states that every non-bipartite graph of order \(n\) with size \(m > \lfloor \frac{ n^2}{ 4} \rfloor\) contains a triangle. \textit{H. Lin} et al. [Comb. Probab. Comput. 30, No. 2, 258--270 (2020; Zbl 1466.05121)] proved a spectral version of Mantel's theorem for given order \(n\). \textit{M. Zhai} and \textit{J. Shu} [Discrete Math. 345, No. 1, Article ID 112630, 10 p. (2022; Zbl 1476.05130)] investigated a spectral version for fixed size \(m\). In this paper, we prove \(Q\)-spectral versions of Mantel's theorem.Two spectral extremal results for graphs with given order and rankhttps://zbmath.org/1517.051072023-09-22T14:21:46.120933Z"Li, Xiuqing"https://zbmath.org/authors/?q=ai:li.xiuqing"Jin, Xian'an"https://zbmath.org/authors/?q=ai:jin.xianan"Shi, Chao"https://zbmath.org/authors/?q=ai:shi.chao"Zheng, Ruiling"https://zbmath.org/authors/?q=ai:zheng.ruilingSummary: The spectral radius and rank of a graph are defined to be the spectral radius and rank of its adjacency matrix, respectively. It is an important problem in spectral extremal graph theory to determine the extremal graph that has the maximum or minimum spectral radius over certain families of graphs. \textit{J. Monsalve} and \textit{J. Rada} [Linear Algebra Appl. 609, 1--11 (2021; Zbl 1458.05152)] obtained the extremal graphs with maximum and minimum spectral radii among all graphs with order \(n\) and rank 4. In this paper, we first determine the extremal graph which attains the maximum spectral radius among all graphs with any given order \(n\) and rank \(r\), and further determine the extremal graph which attains the minimum spectral radius among all graphs with order \(n\) and rank 5.On the Moore-Penrose pseudo-inversion of block symmetric matrices and its application in the graph theoryhttps://zbmath.org/1517.051082023-09-22T14:21:46.120933Z"Pavlíková, Soňa"https://zbmath.org/authors/?q=ai:pavlikova.sona"Ševčovič, Daniel"https://zbmath.org/authors/?q=ai:sevcovic.danielSummary: The purpose of this paper is to analyze the Moore-Penrose pseudo-inversion of symmetric real matrices with application in the graph theory. We introduce a novel concept of positively and negatively pseudo-inverse matrices and graphs. We also give sufficient conditions on the elements of a block symmetric matrix yielding an explicit form of its Moore-Penrose pseudo-inversion. Using the explicit form of the pseudo-inverse matrix we can construct pseudo-inverse graphs for a class of graphs which are constructed from the original graph by adding pendent vertices or pendant paths.On regular graphs equienergetic with their complementshttps://zbmath.org/1517.051092023-09-22T14:21:46.120933Z"Podestá, Ricardo A."https://zbmath.org/authors/?q=ai:podesta.ricardo-a"Videla, Denis E."https://zbmath.org/authors/?q=ai:videla.denis-eAuthors' abstract: We give necessary and sufficient conditions on the parameters of a regular graph \(\Gamma\) (with or without loops) such that \(E(\Gamma) = E(\bar{\Gamma})\). We study complementary equienergetic cubic graphs obtaining classifications up to isomorphisms for connected cubic graphs with single loops (5 non-isospectral pairs) and connected integral cubic graphs without loops (\(\Gamma = K_3\square K_2\) or \(Q_3\)). Then we show that, up to complements, the only bipartite regular graphs equienergetic and nonisospectral with their complements are the crown graphs \(Cr(n)\) or \(C_4\). Next, for the family of strongly regular graphs \(\Gamma\) we characterize all possible parameters \(srg(n, k, e, d)\) such that \(E(\Gamma) = E(\bar{\Gamma})\). Furthermore, using this, we prove that a strongly regular graph is equienergetic to its complement if and only if it is either a conference graph or else it is a pseudo Latin square graph (i.e. has OA parameters). We also characterize all complementary equienergetic pairs of graphs of type \(\mathcal{C}(2)\), \(\mathcal{C}(3)\) and \(\mathcal{C}(5)\) in Cameron's hierarchy (the cases \(\mathcal{C}(1)\) in the nonbipartite case and \(\mathcal{C}(4)\) are still open). Finally, we consider unitary Cayley graphs over rings \(G_R = X(R, R^\ast)\). We show that if \(R\) is a finite Artinian ring with an even number of local factors, then \(G_R\) is complementary equienergetic if and only if \(R = \mathbb{F}_q \times \mathbb{F}_{q^\prime}\) is the product of 2 finite fields.
Reviewer: Hirotake Kurihara (Yamaguchi)Determination of particular double starlike trees by the Laplacian spectrumhttps://zbmath.org/1517.051102023-09-22T14:21:46.120933Z"Stanić, Zoran"https://zbmath.org/authors/?q=ai:stanic.zoranSummary: A double starlike tree is a tree in which exactly two vertices have degree greater than two. In this study we consider double starlike trees obtained by attaching \(p-2\) (for \(p \ge 3\)) pendant vertices at an internal vertex and \(q-2\) (\(q\ge 3\)) pendant vertices at a different internal vertex of a fixed path \(P\). We denote this tree by \(T \cong D(a, b, c, p, q)\), where \(a\), \(b\), \(c\) stand for the numbers of vertices in segments of \(P\) obtained by deleting vertices of degree \(p\) and \(q\). It is known that, depending on parameters, \(T\) may or may not be determined by its Laplacian spectrum. In the latter case we provide the structure of a putative tree with the same Laplacian spectrum. This result implies the known result stating that \(D(1, b, 1, p, q)\) is determined by the Laplacian spectrum and two new results stating the same for \(D(1, b, 2, p, p)\) and \(D(2, b, 2, p, p)\).Distance and adjacency spectra and eigenspaces for three (di)graph lifts: a unified approachhttps://zbmath.org/1517.051112023-09-22T14:21:46.120933Z"Wu, Yongjiang"https://zbmath.org/authors/?q=ai:wu.yongjiang"Zhang, Xiaoqian"https://zbmath.org/authors/?q=ai:zhang.xiaoqian"Feng, Lihua"https://zbmath.org/authors/?q=ai:feng.lihua"Wu, Tingzeng"https://zbmath.org/authors/?q=ai:wu.tingzengSummary: By the induced character theory, we first establish the irreducible decomposition of a permutation representation. Then, using it as a unified tool, we derive a decomposition formula for the distance spectrum and eigenspace of a regular lift of a graph, the adjacency spectra and eigenspaces of a relative lift and a ramified uniform lift of a digraph. The latter results are a complement of \textit{C. Dalfó} et al. [J. Algebr. Comb. 54, No. 2, 651--672 (2021; Zbl 1477.05103)] and \textit{A. Deng} et al. [Eur. J. Comb. 28, No. 4, 1099--1114 (2007; Zbl 1114.05059)].A \(q\)-analogue of distance matrix of block graphshttps://zbmath.org/1517.051122023-09-22T14:21:46.120933Z"Xing, Rundan"https://zbmath.org/authors/?q=ai:xing.rundan"Du, Zhibin"https://zbmath.org/authors/?q=ai:du.zhibinSummary: A \(q\)-analogue of the distance matrix (called \(q\)-distance matrix) of graphs, defined by \textit{W. Yan} and \textit{Y.-N. Yeh} [Adv. Appl. Math. 39, No. 3, 311--321 (2007; Zbl 1129.05029)], is revisited, which is formed from the distance matrix by replacing each nonzero entry \(\alpha\) by \(1 + q + \ldots + q^{\alpha - 1}\) (which would be reduced to \(\alpha\) by setting \(q = 1)\). This concept was also proposed by \textit{R. B. Bapat} et al. [Linear Algebra Appl. 416, No. 2--3, 799--814 (2006; Zbl 1092.05041)]. A graph is called a block graph if every block is a clique (not necessarily of the same order). In this paper, the formula for the inverse of \(q\)-distance matrix of block graphs is presented, which generalizes some classical results about the inverse of distance matrix.Graphical designs and gale dualityhttps://zbmath.org/1517.051132023-09-22T14:21:46.120933Z"Babecki, Catherine"https://zbmath.org/authors/?q=ai:babecki.catherine"Thomas, Rekha R."https://zbmath.org/authors/?q=ai:thomas.rekha-rSummary: A graphical design is a subset of graph vertices such that the weighted averages of certain graph eigenvectors over the design agree with their global averages. We use Gale duality to show that positively weighted graphical designs in regular graphs are in bijection with the faces of a generalized eigenpolytope of the graph. This connection can be used to organize, compute and optimize designs. We illustrate the power of this tool on three families of Cayley graphs -- cocktail party graphs, cycles, and graphs of hypercubes -- by computing or bounding the smallest designs that average all but the last eigenspace in frequency order.Ramsey non-goodness involving bookshttps://zbmath.org/1517.051142023-09-22T14:21:46.120933Z"Fan, Chunchao"https://zbmath.org/authors/?q=ai:fan.chunchao"Lin, Qizhong"https://zbmath.org/authors/?q=ai:lin.qizhongSummary: \textit{S. A. Burr} and \textit{P. Erdős} [J. Graph Theory 7, 39--51 (1983; Zbl 0513.05040)] initiated the study of Ramsey goodness problems. \textit{V. Nikiforov} and \textit{C. C. Rousseau} [Combinatorica 29, No. 2, 227--262 (2009; Zbl 1212.05173)] resolved almost all goodness questions raised by Burr and Erdős [loc. cit.], in which the bounds on the parameters are of tower type since their proofs rely on the regularity lemma. Let \(B_{k,n}\) be the book graph on \(n\) vertices which consists of \(n-k\) copies of \(K_{k+1}\) all sharing a common \(K_k\), and let \(H=K_p(a_1,\dots,a_p)\) be the complete \(p\)-partite graph with parts of sizes \(a_1,\dots,a_p\). Recently, avoiding use of the regularity lemma, \textit{J. Fox} et al. [``Ramsey goodness of books revisited'', Preprint, \url{arXiv:2109.09205}] revisit several Ramsey goodness results involving books. They comment that it would be very interesting to see how far one can push these ideas. In particular, they conjecture that for all integers \(k,p,t\geq 2\), there exists some \(\delta>0\) such that for all \(n\geq 1\), \(1\leq a_1\leq\cdots\leq a_{p-1}\leq t\) and \(a_p\leq\delta n\), we have \(r(H,B_{k,n})=(p-1)(n-1)+d_k(n,K_{a_1,a_2})+1\), where \(d_k(n,K_{a_1,a_2})\) is the maximum \(d\) for which there is an \((n+d-1)\)-vertex \(K_{a_1,a_2}\)-free graph in which at most \(k-1\) vertices have degree less than \(d\). They verify the conjecture when \(a_1=a_2=1\). We disprove the conjecture of Fox et al. [loc. cit.]. Building upon the work of Fox et al., we make a substantial step by showing that for every \(k,p,t\geq 2\), there exists \(\delta>0\) such that the following holds for all large \(n\). Let \(1\leq a_1\leq\dots\leq a_{p-1}\leq t\) and \(a_p\leq\delta n\) be positive integers. If \(a_1=1\), then \(r(H, B_{k,n})\leq(p-1)(n-1)+k(p-1)(a_2-1)+1\). The inequality is tight if \(a_2|(n-1-k)\). Moreover, we prove that for every \(k,a\geq 1\) and \(p\geq 2\), there exists \(\delta>0\) such that for all large \(n\) and \(b\leq\delta\ln n\), \(r(K_p(1,a,b,\dots,b),B_{k,n})=(p-1)(n-1)+k(p-1)(a-1)+1\) if \(a|(n-1-k)\), where the case when \(a=1\) has been proved by Nikiforov and Rousseau [loc. cit.] using the regularity lemma. The bounds on \(1/\delta\) we obtain are not of tower-type since our proofs do not rely on the regularity lemma.Coloring number and on-line Ramsey theory for graphs and hypergraphshttps://zbmath.org/1517.051152023-09-22T14:21:46.120933Z"Kierstead, H. A."https://zbmath.org/authors/?q=ai:kierstead.henry-a"Konjevod, Goran"https://zbmath.org/authors/?q=ai:konjevod.goranSummary: Let \(c\), \(s\), \(t\) be positive integers. The (\(c,s,t\))-\textit{Ramsey game} is played by Builder and Painter. Play begins with an \(s\)-uniform hypergraph \(G_0=(V,E_0)\), where \(E_0=\emptyset\) and \(V\) is determined by Builder. On the \(i\)th round Builder constructs a new edge \(e_i\) (distinct from previous edges) and sets \(G_i =(V,E_i)\), where \(E_i=E_{i -1} \cup\{e_i\}\). Painter responds by coloring \(e_i\) with one of \(c\) colors. Builder wins if Painter eventually creates a monochromatic copy of \(K_s^t\), the complete \(s\)-uniform hypergraph on \(t\) vertices; otherwise Painter wins when she has colored all possible edges.
We extend the definition of coloring number to hypergraphs so that \(\chi (G) \leq \mathrm{col}(G)\) for any hypergraph \(G\) and then show that Builder can win (\(c,s,t\))-Ramsey game while building a hypergraph with coloring number at most \(\mathrm{col}(K_s^t)\). An important step in the proof is the analysis of an auxiliary \textit{survival game} played by Presenter and Chooser. The (\(p,s,t\))-\textit{survival game} begins with an \(s\)-uniform hypergraph \(H_0 = (V,\emptyset)\) with an arbitrary finite number of vertices and no edges. Let \(H_{i -1}=(V_{i -1},E_{i -1})\) be the hypergraph constructed in the first \(i - 1\) rounds. On the \(i\)-th round Presenter plays by presenting a \(p\)-subset \(P_i \subseteq V_{i -1}\) and Chooser responds by choosing an \(s\)-subset \(X_i \subseteq P_i\). The vertices in \(P_i - X_i\) are discarded and the edge \(X_i\) added to \(E_{i -1}\) to form \(E_i\). Presenter wins the survival game if \(H_i\) contains a copy of \(K_s^t\) for some \(i\). We show that for positive integers \(p\), \(s\), \(t\) with \(s \leq p\), Presenter has a winning strategy.Anti-Ramsey number of edge-disjoint rainbow spanning trees in all graphshttps://zbmath.org/1517.051162023-09-22T14:21:46.120933Z"Lu, Linyuan"https://zbmath.org/authors/?q=ai:lu.linyuan"Meier, Andrew"https://zbmath.org/authors/?q=ai:meier.andrew"Wang, Zhiyu"https://zbmath.org/authors/?q=ai:wang.zhiyuSummary: An edge-colored graph \(H\) is called rainbow if every edge of \(H\) receives a different color. Given any host multigraph \(G\), the anti-Ramsey number of \(t\) edge-disjoint rainbow spanning trees in \(G\), denoted by \(r(G,t)\), is defined as the maximum number of colors in an edge-coloring of \(G\) containing no \(t\) edge-disjoint rainbow spanning trees. For any vertex partition \(P\), let \(E(P,G)\) be the set of noncrossing edges in \(G\) with respect to \(P\). In this paper, we determine \(r(G,t)\) for all host multigraphs \(G:r(G,t)=|E(G)|\) if there exists a partition \(P_0\) with \(|E(G)|-|E(P_0,G)|< t(|P_0|-1)\); and \(r(G,t)=\max_{P:|P|\geq 3}\{|E(P,G)|+t(|P|-2)\}\) otherwise. As a corollary, we determine \(r(K_{p,q},t)\) for all values of \(p\), \(q\), \(t\), improving a result of \textit{Y. Jia} et al. [Graphs Comb. 37, No. 2, 409--433 (2021; Zbl 1459.05200)].Hat guessing numbers of strongly degenerate graphshttps://zbmath.org/1517.051172023-09-22T14:21:46.120933Z"Knierim, Charlotte"https://zbmath.org/authors/?q=ai:knierim.charlotte"Martinsson, Anders"https://zbmath.org/authors/?q=ai:martinsson.anders"Steiner, Raphael"https://zbmath.org/authors/?q=ai:steiner.raphael-s|steiner.raphael-m|steiner.raphaelSummary: Assume \(n\) players are placed on \(n\) vertices of a graph \(G\). The following game was introduced by \textit{P. Winkler} [``Games people don't play'', in: Puzzlers' tribute. A feast for the mind. Natick, MA: A K Peters. 301--314 (2002)]: An adversary puts a hat on each player, where each hat has a color out of \(q\) available colors. The players can see the hat of each of their neighbors in \(G\) but cannot see their own hats. Using a predetermined guessing strategy, the players then simultaneously guess the color of their hats. The players win if at least one of them guesses correctly; otherwise, the adversary wins. The largest integer \(q\) such that there is a winning strategy for the players is denoted by \(\mathrm{HG}(G)\), and this is called the hat guessing number of \(G\). Although this game has received much attention in recent years, not much is known about how the hat guessing number relates to other graph parameters. For instance, a natural open question is whether the hat guessing number can be bounded from above in terms of degeneracy. In this paper, we prove that the hat guessing number of a graph can be bounded from above in terms of a related notion, which we call strong degeneracy. We further give an exact characterization of graphs with bounded strong degeneracy. As a consequence, we significantly improve the best known upper bound on the hat guessing number of outerplanar graphs from \(2^{125000}\) to 40 and further derive upper bounds on the hat guessing number for any class of \(K_{2,s}\)-free graphs with bounded expansion, such as the class of \(C_4\)-free planar graphs; more generally, for \(K_{2,s}\)-free graphs with bounded Hadwiger number or without a \(K_t\)-subdivision; and for Erdős-Rényi random graphs with constant average degree.Confining the robber on cographshttps://zbmath.org/1517.051182023-09-22T14:21:46.120933Z"Masjoody, Masood"https://zbmath.org/authors/?q=ai:masjoody.masoodSummary: In a game of Cops and Robbers on graphs, usually the cops' objective is to capture the robber-a situation which the robber wants to avoid invariably. In this paper, we begin with introducing the notions of trapping and confining the robber and discussing their relations with capturing the robber. Our goal is to study the confinement of the robber on graphs that are free of a fixed path as an induced subgraph. We present some necessary conditions for graphs \(G\) not containing the path on \(k\) vertices (referred to as \(P_k\)-free graphs) for some \(k\ge 4\), so that \(k-3\) cops do not have a strategy to capture or confine the robber on \(G\) (Propositions 2.1, 2.3). We then show that for planar cographs and planar \(P_5\)-free graphs the confining cop number is at most one and two, respectively (Corollary 2.4). We also show that the number of vertices of a connected cograph on which one cop does not have a strategy to confine the robber has a tight lower bound of eight. Moreover, we explore the effects of twin operations-which are well known to provide a characterization of cographs-on the number of cops required to capture or confine the robber on cographs. Finally, we pose two conjectures on confining the robber on \(P_5\)-free graphs and the smallest planar graph of confining cop number of three.One-visibility cops and robber on trees: optimal cop-win strategieshttps://zbmath.org/1517.051192023-09-22T14:21:46.120933Z"Yang, Boting"https://zbmath.org/authors/?q=ai:yang.botingSummary: In the one-visibility cops and robber game on a graph, the robber is visible to the cops only when the robber is in the closed neighbourhood of the vertices occupied by the cops. The one-visibility copnumber of a graph is the minimum number of cops required to capture the robber on the graph. In this paper, we investigate the one-visibility cops and robber game on trees. For trees, we introduce a key structure, called road, for characterising optimal cop-win strategies. We give an \(O(n \log n)\) time algorithm to compute an optimal cop-win strategy for a tree with \(n\) vertices. We also establish relations between zero-visibility and one-visibility copnumbers on trees.One-visibility cops and robber on treeshttps://zbmath.org/1517.051202023-09-22T14:21:46.120933Z"Yang, Boting"https://zbmath.org/authors/?q=ai:yang.boting"Akter, Tanzina"https://zbmath.org/authors/?q=ai:akter.tanzinaSummary: The one-visibility cops and robber game is a variation of the classic cops and robber game, where one-visibility means that the information of the robber is known to all cops only when the distance between the robber and at least one cop is at most one. In this paper, we give a lower bound on the one-visibility copnumber of general trees. We present strategies to clear trees according to their structures. We propose a linear-time algorithm for computing the one-visibility copnumber of trees.Induced subgraphs and tree decompositions. IV: (Even hole, diamond, pyramid)-free graphshttps://zbmath.org/1517.051212023-09-22T14:21:46.120933Z"Abrishami, Tara"https://zbmath.org/authors/?q=ai:abrishami.tara"Chudnovsky, Maria"https://zbmath.org/authors/?q=ai:chudnovsky.maria"Hajebi, Sepehr"https://zbmath.org/authors/?q=ai:hajebi.sepehr"Spirkl, Sophie"https://zbmath.org/authors/?q=ai:spirkl.sophie-theresaSummary: A hole in a graph \(G\) is an induced cycle of length at least four, and an even hole is a hole of even length. The diamond is the graph obtained from the complete graph \(K_4\) by removing an edge. A pyramid is a graph consisting of a vertex \(a\) called the apex and a triangle \(\{b_1, b_2, b_3\}\) called the base, and three paths \(P_i\) from \(a\) to \(b_i\) for \(1 \leqslant i \leqslant 3\), all of length at least one, such that for \(i \neq j\), the only edge between \(P_i \setminus \{a\}\) and \(P_j \setminus \{a\}\) is \(b_ib_j\), and at most one of \(P_1, P_2\), and \(P_3\) has length exactly one. For a family \(\mathcal{H}\) of graphs, we say a graph \(G\) is \(\mathcal{H}\)-free if no induced subgraph of \(G\) is isomorphic to a member of \(\mathcal{H}\). \textit{K. Cameron} et al. [Discrete Math. 341, No. 2, 463--473 (2018; Zbl 1376.05132)] proved that (even hole, triangle)-free graphs have treewidth at most five, which motivates studying the treewidth of even-hole-free graphs of larger clique number. \textit{N. L. D. Sintiari} and \textit{N. Trotignon} [J. Graph Theory 97, No. 4, 475--509 (2021; \url{DOI:10.1002/jgt.22666})] provided a construction of (even hole, pyramid, \(K_4)\)-free graphs of arbitrarily large treewidth. Here, we show that for every \(t\), (even hole, pyramid, diamond, \(K_t)\)-free graphs have bounded treewidth. The graphs constructed by Sintiari and Trotignon [loc. cit.] contain diamonds, so our result is sharp in the sense that it is false if we do not exclude diamonds. Our main result is in fact more general, that treewidth is bounded in graphs excluding certain wheels and three-path-configurations, diamonds, and a fixed complete graph. The proof uses ``non-crossing decompositions'' methods similar to those in previous papers in this series. In previous papers, however, bounded degree was a necessary condition to prove bounded treewidth. The result of this paper is the first to use the method of ``non-crossing decompositions'' to prove bounded treewidth in a graph class of unbounded maximum degree.
For Parts I--III see [the first author et al., J. Comb. Theory, Ser. B 157, 144--175 (2022; Zbl 1497.05179); ``Induced subgraphs and tree decompositions. II. Toward walls and their line graphs in graphs of bounded degree'', Preprint, \url{arXiv:2108.01162}; ``Induced subgraphs and tree decompositions. III. Three-path-configurations and logarithmic treewidth'', Preprint, \url{arXiv:2109.01310}].A Torelli theorem for graph isomorphismshttps://zbmath.org/1517.051222023-09-22T14:21:46.120933Z"Griffith, Sarah"https://zbmath.org/authors/?q=ai:griffith.sarahSummary: It is known that isomorphisms of graph Jacobians induce cyclic bijections on the associated graphs. We characterize when such cyclic bijections can be strengthened to graph isomorphisms, in terms of an easily computed divisor. The result refines tools used in algebraic geometry to examine the fibers of the compactified Torelli map.The core of a vertex-transitive complementary prismhttps://zbmath.org/1517.051232023-09-22T14:21:46.120933Z"Orel, Marko"https://zbmath.org/authors/?q=ai:orel.markoSummary: The complementary prism \(\Gamma \bar{\Gamma} \) is obtained from the union of a graph \(\Gamma\) and its complement \(\bar{\Gamma} \) where each pair of identical vertices in \(\Gamma\) and \(\bar{\Gamma} \) is joined by an edge. It generalizes the Petersen graph, which is the complementary prism of the pentagon. The core of a vertex-transitive complementary prism is studied. In particular, it is shown that a vertex-transitive complementary prism \(\Gamma\bar{\Gamma} \) is a core, i.e. all its endomorphisms are automorphisms, whenever \(\Gamma\) is a core or its core is a complete graph.Cut-and-project graphs and other complexeshttps://zbmath.org/1517.051242023-09-22T14:21:46.120933Z"McColm, Gregory L."https://zbmath.org/authors/?q=ai:mccolm.gregory-lSummary: The cut-and-project method may be applied to graphs and complexes, although there are technical difficulties, most notably ``collisions'', i.e. when the images (under the projection) of two disjoint edges or cells intersect. Given a particular periodic structure, a particular projection space, an appropriate ``window'' in the orthogonal complement of that space, the induced substructure within the cartesian product of the window and the projection space is projected to the projection space to produce a ``model structure''. We may use an index space of vectors from the orthogonal complement to move the window around and obtain a ``model system'' of model structures. We adapt the notion of ``general position'' to higher dimensions: the projection space is in ``doubly general position'' with respect to a graph or complex when the projection of that structure maps vertices of that structure injectively into the projection space and the edges or polytopes of that structure injectively so that their dimension is not reduced and disjoint edges and polytopes remain disjoint. We find that if the initial structure was a periodic graph, if the projection space and its complement are in general position with respect to the vertices and edges, and the window is also in general position, then the model system has uncountably many isomorphism classes -- with distinct coordination sequences.Drawing 4-Pfaffian graphs on the torushttps://zbmath.org/1517.051252023-09-22T14:21:46.120933Z"Norine, Serguei"https://zbmath.org/authors/?q=ai:norine.sergueiSummary: We say that a graph \(G\) is \(k\)-Pfaffian if the generating function of its perfect matchings can be expressed as a linear combination of Pfaffians of \(k\) matrices corresponding to orientations of \(G\). We prove that 3-Pfaffian graphs are 1-Pfaffian, 5-Pfaffian graphs are 4-Pfaffian and that a graph is 4-Pfaffian if and only if it can be drawn on the torus (possibly with crossings) so that every perfect matching intersects itself an even number of times. We state conjectures and prove partial results for \(k>5\).Decompositions of quasirandom hypergraphs into hypergraphs of bounded degreehttps://zbmath.org/1517.051262023-09-22T14:21:46.120933Z"Ehard, Stefan"https://zbmath.org/authors/?q=ai:ehard.stefan"Joos, Felix"https://zbmath.org/authors/?q=ai:joos.felixSummary: We prove that any quasirandom uniform hypergraph \(H\) can be approximately decomposed into any collection of bounded degree hypergraphs with almost as many edges. In fact, our results also apply to multipartite hypergraphs and even to the sparse setting when the density of \(H\) quickly tends to 0 in terms of the number of vertices of \(H\). Our results answer and address questions of \textit{J. Kim} et al. [Trans. Am. Math. Soc. 371, No. 7, 4655--4742 (2019; Zbl 1409.05114)]; and \textit{S. Glock} et al. [The existence of designs via iterative absorption: hypergraph \(F\)-designs for arbitrary \(F\). Providence, RI: American Mathematical Society (AMS) (2023; Zbl 1515.05005)] as well as \textit{P. Keevash} [``The existence of designs. II'', Preprint, \url{arXiv:1802.05900}]. The provided approximate decompositions exhibit strong quasirandom properties which are very useful for forthcoming applications. Our results also imply approximate solutions to natural hypergraph versions of long-standing graph decomposition problems, as well as several decomposition results for (quasi)random simplicial complexes into various more elementary simplicial complexes such as triangulations of spheres and other manifolds.On rainbow-free colourings of uniform hypergraphshttps://zbmath.org/1517.051272023-09-22T14:21:46.120933Z"Groot Koerkamp, Ragnar"https://zbmath.org/authors/?q=ai:koerkamp.ragnar-groot"Živný, Stanislav"https://zbmath.org/authors/?q=ai:zivny.stanislavSummary: We study rainbow-free colourings of \(k\)-uniform hypergraphs; that is, colourings that use \(k\) colours but with the property that no hyperedge attains all colours. We show that \(p^\ast = (k - 1)(\ln n) / n\) is the threshold function for the existence of a rainbow-free colouring in a random \(k\)-uniform hypergraph.Hop total Roman domination in graphshttps://zbmath.org/1517.051282023-09-22T14:21:46.120933Z"Abdollahzadeh Ahangar, H."https://zbmath.org/authors/?q=ai:ahangar.hossein-abdollahzadeh"Chellali, M."https://zbmath.org/authors/?q=ai:chellali.mustapha.1"Sheikholeslami, S. M."https://zbmath.org/authors/?q=ai:sheikholeslami.seyed-mahmoud"Soroudi, M."https://zbmath.org/authors/?q=ai:soroudi.marziehSummary: In this article, we initiate a study of hop total Roman domination defined as follows: a \textit{hop total Roman dominating function} (HTRDF) on a graph \(G = (V,E)\) is a function \(f:V \to \{0,1,2\}\) such that for every vertex \(u\) with \(f(u) = 0\) there exists a vertex \(v\) at distance 2 from \(u\) with \(f(v) = 2\) and the subgraph induced by the vertices assigned non-zero values under \(f\) has no isolated vertices. The weight of an HTRDF is the sum of its function values over all vertices, and the hop total Roman domination number \(\gamma_{\mathrm{dtR}}(G)\) equals the minimum weight of an HTRDF on \(G\). We provide several properties on the hop total Roman domination number. More precisely, we show that the decision problem corresponding to the hop total Roman domination problem is NP-complete for bipartite graphs, and we determine the exact value of \(\gamma_{\mathrm{htR}}(G)\) for paths and cycles. Moreover, we characterize all connected graphs \(G\) of order \(n\) with \(\gamma_{\mathrm{htR}}(G) \in \{2,3,4,n\}\). Finally, we show that for every tree \(T\) of diameter at least \(3\), \(\gamma_{\mathrm{htR}}(T) \geq \gamma_{\mathrm{ht}}(T)+2\) where \(\gamma_{\mathrm{ht}}(T)\) is the hop total domination number.Choice functionshttps://zbmath.org/1517.051292023-09-22T14:21:46.120933Z"Aharoni, Ron"https://zbmath.org/authors/?q=ai:aharoni.ron"Briggs, Joseph"https://zbmath.org/authors/?q=ai:briggs.josephSummary: This is a survey paper on rainbow sets (another name for ``choice functions''). The main theme is the distinction between two types of choice functions: those having a large (in the sense of belonging to some specified filter, namely closed up set of sets) image, and those that have a large domain and small image, where ``smallness'' means belonging to some specified complex (a closed-down set). The paper contains some new results: (1) theorems on scrambled versions, in which the sets are re-shuffled before choosing the rainbow set, and (2) results on weighted and cooperative versions -- to be defined below.On first Zagreb energy of total dominating set in graphshttps://zbmath.org/1517.051302023-09-22T14:21:46.120933Z"Bilar, V. T."https://zbmath.org/authors/?q=ai:bilar.v-t"Dagondon, S. C."https://zbmath.org/authors/?q=ai:dagondon.s-cSummary: Let \(G\) be a simple graph of order \(n\) with vertex set \(V(G)\) and edge set \(E(G)\). Let \(\mathrm{MTD}\) be the minimum total dominating set of \(G\) and \(|\mathrm{MTD}| = \gamma_t(G)\) the total domination number. The first Zagreb matrix of minimum total dominating set of \(G\) is the \(n \times n\) matrix defined by \(\operatorname{Z}_{\mathrm{MTD}}(G) = (z_{ij})\) where
\[
\operatorname{Z}_{\mathrm{MTD}}(G)=
\begin{cases}
d_i + d_j, &\text{if } v_i \text{ and } v_j \text{ are adjacent}\\
1, &\text{if } i = j\text{ and }v_i\in \mathrm{MTD}\\
0, &\text{otherwise}.
\end{cases}
\]
where \(d_i\) and \(d_j\) are the degree of the vertex \(v_i\) and \(v_j\) (\(1 \leq i < j \leq n\)), respectively and \(v_1, v_2, \cdots, v_n\) are the vertices of the \(n\times n\) graph \(G\). The first Zagreb energy of total dominating set of \(G\), denoted by \(\operatorname{ZE}_{TD}(G)\), is the summation of the \textit{eigenvalues} of \(G\) where the \textit{eigenvalues} of \(G\) are the zeros of the characteristic polynomial of \(\operatorname{Z}_{\mathrm{MTD}}(G)\). In this study, we determine the exact values or bounds of \(\operatorname{ZE}_{\mathrm{TD}}(G)\) of some special graphs. Moreover, we also provide some characterizations for \(\operatorname{ZE}_{\mathrm{TD}}(G)\) and its relationship with Zagreb energy \(\operatorname{ZE}(G)\).Eternal distance-\(k\) domination on graphshttps://zbmath.org/1517.051312023-09-22T14:21:46.120933Z"Cox, D."https://zbmath.org/authors/?q=ai:cox.david-a|cox.d-p-g|cox.daniel-j|cox.dennis-d|cox.darrell|cox.donald-c|cox.david-eric|cox.dana|cox.david-n|cox.david-r|cox.david-d|cox.david-o|cox.danielle"Meger, E."https://zbmath.org/authors/?q=ai:meger.erin"Messinger, M. E."https://zbmath.org/authors/?q=ai:messinger.margaret-ellenSummary: Eternal domination is a dynamic process by which a graph is protected from an infinite sequence of vertex intrusions. In eternal distance-\(k\) domination, guards initially occupy the vertices of a distance-\(k\) dominating set. After a vertex is attacked, guards ``defend'' by each moving up to distance \(k\) to form a distance-\(k\) dominating set, such that some guard occupies the attacked vertex. The eternal distance-\(k\) domination number of a graph is the minimum number of guards needed to defend against any sequence of attacks. The process is well-studied for the situation where \(k=1\). We introduce eternal distance-\(k\) domination for \(k > 1\). Determining whether a given set is an eternal distance-\(k\) domination set is in EXP, and in this paper we provide a number of results for paths and cycles, and relate this parameter to graph powers and domination in general. For trees we use decomposition arguments to bound the eternal distance-\(k\) domination numbers, and solve the problem entirely in the case of perfect \(m\)-ary trees.Quasi-total Roman reinforcement in graphshttps://zbmath.org/1517.051322023-09-22T14:21:46.120933Z"Ebrahimi, N."https://zbmath.org/authors/?q=ai:ebrahimi.nafiseh"Amjadi, J."https://zbmath.org/authors/?q=ai:amjadi.jafar"Chellali, M."https://zbmath.org/authors/?q=ai:chellali.mustapha.1"Sheikholeslami, S. M."https://zbmath.org/authors/?q=ai:sheikholeslami.seyed-mahmoudSummary: A quasi-total Roman dominating function (QTRD-function) on \(G = (V,E)\) is a function \(f:V \to \{0,1,2\}\) such that (i) every vertex \(x\) for which \(f(x) = 0\) is adjacent to at least one vertex \(v\) for which \(f(v) = 2\), and (ii) if \(x\) is an isolated vertex in the subgraph induced by the set of vertices with non-zero values, then \(f(x) = 1\). The weight of a QTRD-function is the sum of its function values over the whole set of vertices, and the quasi-total Roman domination number is the minimum weight of a QTRD-function on \(G\). The quasi-total Roman reinforcement number \(r_{qtR}(G)\) of a graph \(G\) is the minimum number of edges that have to be added to \(G\) in order to decrease the quasi-total Roman domination number. In this paper, we initiate the study of quasi-total Roman reinforcement in graphs. We first show that the decision problem associated with the quasi-total Roman reinforcement problem is NP-hard even when restricted to bipartite graphs. Then basic properties of the quasi-total Roman reinforcement number are provided. Finally, some sharp bounds for \(r_{qtR}(G)\) are also presented.Dominance complexes and vertex cover numbers of graphshttps://zbmath.org/1517.051332023-09-22T14:21:46.120933Z"Matsushita, Takahiro"https://zbmath.org/authors/?q=ai:matsushita.takahiroSummary: The dominance complex \(D(G)\) of a simple graph \(G = (V,E)\) is the simplicial complex consisting of the subsets of \(V\) whose complements are dominating. We show that the connectivity of \(D(G)\) plus 2 is a lower bound for the vertex cover number \(\tau (G)\) of \(G\).\(k\)-efficient domination: algorithmic perspectivehttps://zbmath.org/1517.051342023-09-22T14:21:46.120933Z"Meybodi, Mohsen Alambardar"https://zbmath.org/authors/?q=ai:meybodi.mohsen-alambardarSummary: A set \(D\subseteq V\) of a graph \(G=(V,E)\) is called an efficient dominating set of \(G\) if every vertex \(v\) has exactly one neighbor in \(D\), in other words, the vertex set \(V\) is partitioned to some circles with radius one such that the vertices in \(D\) are the centers of partitions. A generalization of this concept, introduced by \textit{M. Chellali} et al. [Commun. Comb. Optim. 4, No. 2, 109--122 (2019; Zbl 1438.05190)], is called \(k\)-efficient dominating set that briefly partitions the vertices of graph with different radiuses. It leads to a partition set \(\{U_1, U_2,\dots, U_t\}\) such that each \(U_i\) consists a center vertex \(u_i\) and all the vertices in distance \(d_i\), where \(d_i\in\{0,1,\dots,k\}\). In other words, there exist the dominators with various dominating powers. The problem of finding minimum set \(\{ u_1, u_2,\dots, u_t\}\) is called the minimum \(k\)-efficient domination problem. Given a positive integer \(S\) and a graph \(G=(V,E)\), the \(k\)-efficient Domination Decision problem is to decide whether \(G\) has a \(k\)-efficient dominating set of cardinality at most \(S\). The \(k\)-efficient Domination Decision problem is known to be NP-complete even for bipartite graphs [loc. cit.]. Clearly, every graph has a \(k\)-efficient dominating set but it is not correct for efficient dominating set. In this paper, we study the following:
\begin{itemize}
\item[\(\bullet\)] \(k\)-efficient domination problem set is NP-complete even in chordal graphs.
\item[\(\bullet\)] A polynomial-time algorithm for \(k\)-efficient domination in trees.
\item[\(\bullet\)] \(k\)-efficient domination on sparse graphs from the parametrized complexity perspective. In particular, we show that it is \(W[1]\)-hard on \(d\)-degenerate graphs while the original dominating set has Fixed Parameter Tractable (FPT) algorithm on \(d\)-degenerate graphs.
\item[\(\bullet\)] \(k\)-efficient domination on nowhere-dense graphs is FPT.
\end{itemize}Secure equitability in graphshttps://zbmath.org/1517.051352023-09-22T14:21:46.120933Z"Muthusubramanian, L."https://zbmath.org/authors/?q=ai:muthusubramanian.l"Sundareswaran, R."https://zbmath.org/authors/?q=ai:sundareswaran.r"Swaminathan, V."https://zbmath.org/authors/?q=ai:swaminathan.venkataSummary: In secure domination
[\textit{A. P. Burger} et al., Quaest. Math. 31, No. 2, 163--171 (2008; Zbl 1154.05048);
\textit{E. J. Cockayne}, Discrete Math. 307, No. 1, 12--17 (2007; Zbl 1233.05143);
Bull. Inst. Comb. Appl. 39, 87--100 (2003; Zbl 1051.05065);
\textit{C. M. Mynhardt} et al., Util. Math. 67, 255--267 (2005; Zbl 1071.05058)],
a vertex outside has the chance of coming inside the dominating set by replacing an element of the set without affecting domination. This idea is combined with equitability. Secure equitable dominating set is introduced and studied. Other concepts like independence and rigid security are also studied in this paper.Dominating induced matching in some subclasses of bipartite graphshttps://zbmath.org/1517.051362023-09-22T14:21:46.120933Z"Panda, B. S."https://zbmath.org/authors/?q=ai:panda.bhawani-sankar"Chaudhary, Juhi"https://zbmath.org/authors/?q=ai:chaudhary.juhiSummary: A subset \(M \subseteq E\) of edges of a graph \(G = (V, E)\) is called a matching if no two edges of \(M\) share a common vertex. An edge \(e \in E\) is said to dominate itself and all other edges adjacent to it. A matching \(M\) in a graph \(G = (V, E)\) is called a dominating induced matching (d.i.m.) if every edge of \(G\) is dominated by edges of \(M\) exactly once. The dominating induced matching decide (\textsc{DIM-Decide}) problem asks whether a graph \(G\) contains a dominating induced matching. The dominating induced matching (\textsc{DIM}) problem asks to compute a dominating induced matching (d.i.m.) in a graph \(G\) that admits a dominating induced matching. The \textsc{DIM-Decide} problem is known to be NP-complete for general graphs as well as for bipartite graphs. In this paper, we strengthen the NP-completeness result of the \textsc{DIM-Decide} problem by showing that this problem remains NP-complete for perfect elimination bipartite graphs, a proper subclass of bipartite graphs. On the positive side, we characterize the class of star-convex bipartite graphs admitting a d.i.m. This characterization leads to a linear time algorithm to test whether a star-convex bipartite graph admits a d.i.m. and, if so, constructs a d.i.m. in such a star-convex bipartite graph in linear time. We also propose polynomial time algorithms to construct a d.i.m. in long-\(k\)-star-convex bipartite graphs as well as in circular-convex bipartite graphs if the input graph admits a d.i.m.A substructure based lower bound for eternal vertex cover numberhttps://zbmath.org/1517.051372023-09-22T14:21:46.120933Z"Babu, Jasine"https://zbmath.org/authors/?q=ai:babu.jasine"Prabhakaran, Veena"https://zbmath.org/authors/?q=ai:prabhakaran.veena"Sharma, Arko"https://zbmath.org/authors/?q=ai:sharma.arkoSummary: The eternal vertex cover (EVC) problem is to compute the minimum number of guards to be placed on the vertices of a graph so that any sequence of attacks on its edges can be defended by dynamically reconfiguring the guards. The problem is NP-hard in general and polynomial time algorithms are unknown even for simple graph classes like cactus graphs and bipartite graphs. A major difficulty is that only few lower bounds, other than the trivial lower bound of vertex cover, is known in general and the known bounds are too weak to yield useful results even for the graph classes mentioned above. We introduce the notion of substructure property in the context of the EVC problem and derive a new lower bounding technique for the problem based on the property. We apply the technique to cactus graphs and chordal graphs and obtain new algorithms for solving the eternal vertex cover problem in linear time for cactus graphs and quadratic time for a family of graphs that includes all chordal graphs and cactus graphs.Perfect matching complexes of honeycomb graphshttps://zbmath.org/1517.051382023-09-22T14:21:46.120933Z"Bayer, Margaret"https://zbmath.org/authors/?q=ai:bayer.margaret-m"Jelić Milutinović, Marija"https://zbmath.org/authors/?q=ai:milutinovic.marija-jelic"Vega, Julianne"https://zbmath.org/authors/?q=ai:vega.julianneSummary: The perfect matching complex of a graph is the simplicial complex on the edge set of the graph with facets corresponding to perfect matchings of the graph. This paper studies the perfect matching complexes, \(\mathcal{M}_p(H_{k \times m\times n})\), of honeycomb graphs. For \(k = 1\), \(\mathcal{M}_p(H_{1\times m\times n})\) is contractible unless \(n\geqslant m=2\), in which case it is homotopy equivalent to the \((n-1)\)-sphere. Also, \(\mathcal{M}_p(H_{2\times 2\times 2})\) is homotopy equivalent to the wedge of two 3-spheres. The proofs use discrete Morse theory.Balancing permuted copies of multigraphs and integer matriceshttps://zbmath.org/1517.051392023-09-22T14:21:46.120933Z"del Valle, Coen"https://zbmath.org/authors/?q=ai:del-valle.coen"Dukes, Peter J."https://zbmath.org/authors/?q=ai:dukes.peter-jamesSummary: Given a square matrix \(A\) over the integers, we consider the \(\mathbb{Z} \)-module \(M_A\) generated by the set of all matrices that are permutation-similar to \(A\). Motivated by analogous problems on signed graph decompositions and block designs, we are interested in the completely symmetric matrices \(a I + b J\) belonging to \(M_A\). We give a relatively fast method to compute a generator for such matrices, avoiding the need for a very large canonical form over \(\mathbb{Z} \). Several special cases are considered. In particular, the problem for symmetric matrices answers a question of \textit{S. M. Cioabă} and \textit{P. J. Cameron} [Am. Math. Mon. 122, No. 10, 972--981 (2015; Zbl 1338.05175)] on determining the eventual period for integers \(\lambda\) such that the \(\lambda \)-fold complete graph \(\lambda K_n\) has an edge-decomposition into a given (multi)graph.Lower bounds for maximum weighted cuthttps://zbmath.org/1517.051402023-09-22T14:21:46.120933Z"Gutin, Gregory"https://zbmath.org/authors/?q=ai:gutin.gregory-z"Yeo, Anders"https://zbmath.org/authors/?q=ai:yeo.andersSummary: While there have been many results on lower bounds for Max Cut in unweighted graphs, the only lower bound for noninteger weights is that by \textit{S. Poljak} and \textit{D. Turzík} [Discrete Math. 58, 99--104 (1986; Zbl 0585.05032)]. In this paper, we launch an extensive study of lower bounds for Max Cut in weighted graphs. We introduce a new approach for obtaining lower bounds for Weighted Max Cut. Using it, the probabilistic method, Vizing's chromatic index theorem, and other tools, we obtain several lower bounds for arbitrary weighted graphs, weighted graphs of bounded girth, and triangle-free weighted graphs. We pose conjectures and open questions.On Petrie cycle and Petrie tour partitions of 3- and 4-regular plane graphshttps://zbmath.org/1517.051412023-09-22T14:21:46.120933Z"He, Xin"https://zbmath.org/authors/?q=ai:he.xin"Zhang, Huaming"https://zbmath.org/authors/?q=ai:zhang.huaming"Han, Yijie"https://zbmath.org/authors/?q=ai:han.yijieSummary: Given a plane graph \(G=(V,E)\), a Petrie tour of \(G\) is a tour \(P\) of \(G\) that alternately turns left and right at each step. A Petrie tour partition of \(G\) is a collection \({\mathscr P}=\{P_1, \ldots, P_q\}\) of Petrie tours so that each edge of \(G\) is in exactly one tour \(P_i \in{\mathscr P} \). A Petrie tour \(P\) is called a Petrie cycle if all its vertices are distinct. A Petrie cycle partition of \(G\) is a collection \({\mathscr C}=\{C_1, \ldots, C_p\}\) of Petrie cycles so that each vertex of \(G\) is in exactly one cycle \(C_i \in\mathscr{C}\). In this paper, we study the properties of 3-regular plane graphs that have Petrie cycle partitions and 4-regular plane multi-graphs that have Petrie tour partitions. Given a 4-regular plane multi-graph \(G=(V, E)\), a 3-regularization of \(G\) is a 3-regular plane graph \(G_3\) obtained from \(G\) by splitting every vertex \(v\in V\) into two degree-3 vertices. \(G\) is called Petrie partitionable if it has a 3-regularization that has a Petrie cycle partition. The general version of this problem is motivated by a data compression method, tristrip, used in computer graphics. In this paper, we present a simple characterization of Petrie partitionable graphs and show that the problem of determining if \(G\) is Petrie partitionable is NP-complete.The superstar packing problemhttps://zbmath.org/1517.051422023-09-22T14:21:46.120933Z"Janata, Marek"https://zbmath.org/authors/?q=ai:janata.marek"Szabó, Jácint"https://zbmath.org/authors/?q=ai:szabo.jacintSummary: \textit{P. Hell} and \textit{D. G. Kirkpatrick} [SIAM J. Algebraic Discrete Methods 7, 199--209 (1986; Zbl 0597.05050)] proved that in an undirected graph, a maximum size packing by a set of non-singleton stars can be found in polynomial time if this star-set is of the form \(\{S_1, S_2, \dots, S_k\}\) for some \(k \in \mathbb{Z}_+\) (\(S_i\) is the star with \(i\) leaves), and it is NP-hard otherwise. This may raise the question whether it is possible to enlarge a set of stars not of the form \(\{S_1, S_2, \dots, S_k\}\) by other non-star graphs to get a polynomially solvable graph packing problem. This paper shows such families of depth 2 trees. We show two approaches to this problem, a polynomial alternating forest algorithm, which implies a Berge-Tutte type min-max theorem, and a reduction to the degree constrained subgraph problem of Lovász.The minimum degree threshold for perfect graph packingshttps://zbmath.org/1517.051432023-09-22T14:21:46.120933Z"Kühn, Daniela"https://zbmath.org/authors/?q=ai:kuhn.daniela"Osthus, Deryk"https://zbmath.org/authors/?q=ai:osthus.derykSummary: Let \(H\) be any graph. We determine up to an additive constant the minimum degree of a graph \(G\) which ensures that \(G\) has a perfect \(H\)-packing (also called an \(H\)-factor). More precisely, let \(\delta(H,n)\) denote the smallest integer \(k\) such that every graph \(G\) whose order \(n\) is divisible by \(|H|\) and with \(\delta (G) \geq k\) contains a perfect \(H\)-packing. We show that
\[
\delta (H,n) = \left(1 - \frac{1}{\chi^* (H)} \right)n + O(1).
\]
The value of \(\chi^*(H)\) depends on the relative sizes of the colour classes in the optimal colourings of \(H\) and satisfies \(\chi (H)-1<\chi^*(H) \leq \chi (H)\).Minimal instances with no weakly stable matching for three-sided problem with cyclic incomplete preferenceshttps://zbmath.org/1517.051442023-09-22T14:21:46.120933Z"Lerner, Eduard Yu."https://zbmath.org/authors/?q=ai:lerner.eduard-yulevich"Lerner, Regina E."https://zbmath.org/authors/?q=ai:lerner.regina-eSummary: Given \(n\) men, \(n\) women, and \(n\) dogs, each man has an incomplete preference list of women, each woman has an incomplete preference list of dogs, and each dog has an incomplete preference list of men. We understand a family as a triple consisting of one man, one woman, and one dog such that the dog belongs to the preference list of the woman, who, in turn, belongs to the preference list of the man, while the latter belongs to the preference list of the dog. We understand a matching as a collection of nonintersecting families (some agents, possibly, remain single). A matching is said to be nonstable, if one can find a man, a woman, and a dog who do not live together currently but each of them would become ``happier'' if they do. Otherwise, the matching is said to be stable (a weakly stable matching). We give an example of this problem for \(n=3\) where no stable matching exists. Moreover, we prove the absence of such an example for \(n<3\). Such an example was known earlier only for \(n=6\) [\textit{P. Biró} and \textit{E. McDermid}, Algorithmica 58, No. 1, 5--18 (2010; Zbl 1205.68257)]. The constructed examples also allow one to halve the size of the recently constructed analogous example for complete preference lists [\textit{C.-K. Lam} and \textit{C. G. Plaxton}, Lect. Notes Comput. Sci. 11801, 329--342 (2019; Zbl 1431.91259)].Two-disjoint-cycle-cover vertex bipancyclicity of bipartite hypercube-like networkshttps://zbmath.org/1517.051452023-09-22T14:21:46.120933Z"Niu, Ruichao"https://zbmath.org/authors/?q=ai:niu.ruichao"Zhou, Shujie"https://zbmath.org/authors/?q=ai:zhou.shujie"Xu, Min"https://zbmath.org/authors/?q=ai:xu.minSummary: Let \(r_1\), \(r_2\) be two integers such that \(r_2 \geq r_1 \geq 0\). A bipartite graph \(G\) is two-disjoint-cycle-cover vertex \([ r_1, r_2]\)-bipancyclic (2-DCC vertex \([ r_1, r_2]\)-bipancyclic for short) if for any two vertices \(u, v \in V(G)\) and any even integer \(\ell\) satisfying \(r_1 \leq \ell \leq r_2\), there exist two vertex-disjoint cycles \(C_1\) and \(C_2\) in \(G\) with \(| V( C_1) | = \ell\) and \(| V( C_2) | = | V(G) | - \ell\) such that \(u \in V( C_1)\) and \(v \in V( C_2)\). In this paper, we study the 2-DCC vertex bipancyclicity of the \(n\)-dimensional bipartite hypercube-like network, which is one class of hypercube-generalized networks. As a consequence, we show that an \(n\)-dimensional bipartite hypercube-like network is 2-DCC vertex \([4, 2^{n - 1}]\)-bipancyclic for \(n \geq 3\). In particular, it provides an application that \(n\)-dimensional hypercube and bicube are also 2-DCC vertex \([4, 2^{n - 1}]\)-bipancyclic for \(n \geq 3\).A simple combinatorial algorithm for restricted 2-matchings in subcubic graphs -- via half-edgeshttps://zbmath.org/1517.051462023-09-22T14:21:46.120933Z"Paluch, Katarzyna"https://zbmath.org/authors/?q=ai:paluch.katarzyna-e"Wasylkiewicz, Mateusz"https://zbmath.org/authors/?q=ai:wasylkiewicz.mateuszSummary: We consider three variants of the problem of finding a maximum weight restricted 2-matching in a subcubic graph \(G\). (A 2-matching is any subset of the edges such that each vertex is incident to at most two of its edges.) Depending on the variant a restricted 2-matching means a 2-matching that is either triangle-free or square-free or both triangle- and square-free. Since computing a maximum weight square-free 2-matching in a subcubic graph is NP-hard, in the second and third variant we additionally assume that the edge-weights are vertex-induced on each square. While there exist polynomial time algorithms for the first two types of 2-matchings, they are quite complicated or use advanced methodology. For each of the three problems we present a simple reduction to the computation of a maximum weight \(b\)-matching. The reduction is conducted with the aid of half-edges. A half-edge of edge \(e\) is, informally speaking, a half of \(e\) containing exactly one of its endpoints. For a subset of triangles and/or squares of \(G\), we replace each edge of such a triangle/square with two half-edges. Two half-edges of one edge \(e\) of weight \(w(e)\) may get different weights, not necessarily equal to \(\frac{1}{2}w(e)\). In the metric setting when the edge weights satisfy the triangle inequality, this has a geometric interpretation connected to how an incircle partitions the edges of a triangle. Our algorithms are additionally faster than those known before. The running time of each of them is \(O(n^2\log n)\), where \(n\) denotes the number of vertices in the graph.Recoloring subgraphs of \(K_{2n}\) for sports schedulinghttps://zbmath.org/1517.051472023-09-22T14:21:46.120933Z"Urrutia, Sebastián"https://zbmath.org/authors/?q=ai:urrutia.sebastian-alberto"de Werra, Dominique"https://zbmath.org/authors/?q=ai:de-werra.dominique"Januario, Tiago"https://zbmath.org/authors/?q=ai:januario.tiagoSummary: The exploration of one-factorizations of complete graphs is the foundation of some classical sports scheduling problems. One has to traverse the landscape of such one-factorizations by moving from one of those to a so-called neighbor one-factorization. This approach amounts to modifying locally the coloring associated with a one-factorization. We consider some particular types of modifications and describe various constructions which give one-factorizations which may be modified or not by these techniques. Among those are recoloring of bichromatic cycles, altering of optimally colored subcliques of even size, or recoloring of chordless lanterns.Balanced subdivisions of a large clique in graphs with high average degreehttps://zbmath.org/1517.051482023-09-22T14:21:46.120933Z"Wang, Yan"https://zbmath.org/authors/?q=ai:wang.yan.32|wang.yan.14|wang.yan.18|wang.yan.52|wang.yan.7|wang.yan|wang.yan.5|wang.yan.3|wang.yan.58|wang.yan.36|wang.yan.47|wang.yan.34|wang.yan.59|wang.yan.35|wang.yan.28|wang.yan.49|wang.yan.2|wang.yan.42|wang.yan.1|wang.yan.41|wang.yan.19|wang.yan.57|wang.yan.27|wang.yan.53Summary: \textit{C. Thomassen} [J. Graph Theory 8, 23--28 (1984; Zbl 0535.05051)] conjectured that for every constant \(k\in\mathbb{N}\), there exists \(d\) such that every graph with average degree at least \(d\) contains a balanced subdivision of a complete graph on \(k\) vertices, i.e., a subdivision in which each edge is subdivided the same number of times. Recently, \textit{H. Liu} and \textit{R. Montgomery} [J. Lond. Math. Soc., II. Ser. 95, No. 1, 203--222 (2017; Zbl 1370.05103)] confirmed Thomassen's conjecture. We show that for every constant \(0<c<1/2\), every graph with average degree at least \(d\) contains a balanced subdivision of a complete graph of size at least \(\Omega (d^c)\). Note that this bound is almost optimal. Moreover, we show that every sparse expander with minimum degree at least \(d\) contains a balanced subdivision of a complete graph of size at least \(\Omega (d)\).Inertia indices of a complex unit gain graph in terms of matching numberhttps://zbmath.org/1517.051492023-09-22T14:21:46.120933Z"Wu, Qi"https://zbmath.org/authors/?q=ai:wu.qi"Lu, Yong"https://zbmath.org/authors/?q=ai:lu.yongSummary: A complex unit gain graph is a triple \(\Phi = (G, \mathbb{T}, \varphi)\) (or \(G^\varphi\) for short) consisting of a simple graph \(G\), as the underlying graph of \(G^\varphi\), the set of unit complex numbers \(\mathbb{T} = \{z \in \mathbb{C} : |z| = 1\}\) and a gain function \(\varphi : \overrightarrow{E} \to \mathbb{T}\) such that \(\varphi(e_{i,j}) = \varphi(e_{j,i})^{-1}\). Let \(A(G^\varphi)\) be the adjacency matrix of \(G^\varphi\). In this paper, we prove that
\[
\begin{aligned}
m(G) - c(G) \leq p(G^\varphi) \leq m(G) + c(G),\\
m(G) - c(G) \leq n(G^\varphi) \leq m(G) + c(G),
\end{aligned}
\]
where \(p(G^\varphi)\), \(n(G^\varphi)\), \(m(G)\) and \(c(G)\) are the number of positive eigenvalues of \(A(G^\varphi)\), the number of negative eigenvalues of \(A(G^\varphi)\), the matching number and the cyclomatic number of \(G\), respectively. Furthermore, we characterize the graphs which attain the upper bounds and the lower bounds, respectively.Isolated toughness and path-factor uniform graphs. II.https://zbmath.org/1517.051502023-09-22T14:21:46.120933Z"Zhou, Sizhong"https://zbmath.org/authors/?q=ai:zhou.sizhong"Sun, Zhiren"https://zbmath.org/authors/?q=ai:sun.zhiren"Bian, Qiuxiang"https://zbmath.org/authors/?q=ai:bian.qiuxiangSummary: A spanning subgraph \(F\) of \(G\) is called a path-factor if each component of \(F\) is a path. A \(P_{\geq k}\)-factor of \(G\) means a path-factor such that each component is a path with at least \(k\) vertices, where \(k \geq 2\) is an integer. A graph \(G\) is called a \(P_{\geq k}\)-factor covered graph if for each \(e\in E(G)\), \(G\) has a \(P_{\geq k}\)-factor covering \(e\). A graph \(G\) is called a \(P_{\geq k}\)-factor uniform graph if for any two different edges \(e_1,e_2\in E(G)\), \(G\) has a \(P_{\geq k}\)-factor covering \(e_1\) and avoiding \(e_2\). In other word, a graph \(G\) is called a \(P_{\geq k}\)-factor uniform graph if for any \(e\in E(G)\), the graph \(G-e\) is a \(P_{\geq k}\)-factor covered graph. In this article, we demonstrate that (i) an \((r+3)\)-edge-connected graph \(G\) is a \(P_{\geq 2}\)-factor uniform graph if its isolated toughness \(I(G) > \frac{r+3}{2r+3}\), where \(r\) is a nonnegative integer; (ii) an \((r+3)\)-edge-connected graph \(G\) is a \(P_{\geq 3}\)-factor uniform graph if its isolated toughness \(I(G) > \frac{3r+6}{2r+3}\), where \(r\) is a nonnegative integer. Furthermore, we claim that these conditions on isolated toughness and edge-connectivity in our main results are best possible in some sense.
For Part I see [\textit{S. Zhou} et al., RAIRO, Oper. Res. 55, No. 3, 1279--1290 (2021; Zbl 1468.05243)].A note on fold thickness of graphshttps://zbmath.org/1517.051512023-09-22T14:21:46.120933Z"Reji, T."https://zbmath.org/authors/?q=ai:reji.t"Vaishnavi, S."https://zbmath.org/authors/?q=ai:vaishnavi.s"Campeña, Francis Joseph H."https://zbmath.org/authors/?q=ai:campena.francis-joseph-hSummary: A 1-fold of \(G\) is the graph \(G^0\) obtained from a graph \(G\) by identifying two nonadjacent vertices in \(G\) having at least one common neighbor and reducing the resulting multiple edges to simple edges. A uniform \(k\)-folding of a graph \(G\) is a sequence of graphs \(G = G_0, G_1, G_2,\dots, G_k\), where \(G_{i+1}\) is a 1-fold of \(G_i\) for \(i = 0, 1, 2,\dots, k-1\) such that all graphs in the sequence are singular or all of them are nonsingular. The largest \(k\) for which there exists a uniform \(k\)- folding of \(G\) is called fold thickness of \(G\) and this concept was first introduced in [\textit{F. J. H. Campeña} and \textit{S. V. Gervacio}, Arab. J. Math. 9, No. 2, 345--355 (2020; Zbl 1443.05162)]. In this paper, we determine fold thickness of corona product graph \(G \odot \overline{K_m}, G \odot_S \overline{K_m}\) and graph join \(G + \overline{K_m}\).On graceful antimagic graphshttps://zbmath.org/1517.051522023-09-22T14:21:46.120933Z"Ahmed, Mohammed Ali"https://zbmath.org/authors/?q=ai:ahmed.mohammed-ali"Semaničová-Feňovčíková, Andrea"https://zbmath.org/authors/?q=ai:semanicova-fenovcikova.andrea"Bača, Martin"https://zbmath.org/authors/?q=ai:baca.martin"Babujee, J. Baskar"https://zbmath.org/authors/?q=ai:babujee.jayapal-baskar|babujee.j-baskar"Shobana, Loganathan"https://zbmath.org/authors/?q=ai:shobana.loganathanGraceful labeling and antimagic labeling are two significant topics in the domain of graph labelings, with outstanding conjectures which still remain unsolved. In this paper, the authors combine these two concepts to define a new labeling, called graceful antimagic labelings. Graceful antimagicness of some families of trees, cycles, and nearly complete graphs is established. Several conjectures and open problems for further investigation are posed. Also, a graceful local antimagic labeling that induces a proper vertex coloring is proposed and this concept has wide scope for further investigation.
Reviewer: S. Arumugam (Krishnankoil)Optimal radio labellings of block graphs and line graphs of treeshttps://zbmath.org/1517.051532023-09-22T14:21:46.120933Z"Bantva, Devsi"https://zbmath.org/authors/?q=ai:bantva.devsi-d"Liu, Daphne Der-Fen"https://zbmath.org/authors/?q=ai:liu.daphne-der-fenSummary: A radio labeling of a graph \(G\) is a mapping \(f : V(G) \rightarrow \{0, 1, 2,\dots\}\) such that \(| f(u) - f(v) | \geqslant d(G) + 1 - d(u, v)\) holds for every pair of vertices \(u\) and \(v\), where \(d(G)\) is the diameter of \(G\) and \(d(u, v)\) is the distance between \(u\) and \(v\) in \(G\). The radio number of \(G\), denoted by \(r n(G)\), is the smallest \(t\) such that \(G\) admits a radio labeling with \(t = \max \{| f(v) - f(u) | : v, u \in V(G) \}\). A block graph is a graph such that each block (induced maximal 2-connected subgraph) is a complete graph. In this paper, a lower bound for the radio number of block graphs is established. The block graph which achieves this bound is called a lower bound block graph. We prove three necessary and sufficient conditions for lower bound block graphs. Moreover, we give three sufficient conditions for a graph to be a lower bound block graph. Using these results, we present several families of lower bound block graphs, including the level-wise regular block graphs and the extended star of blocks. The line graph of a graph \(G(V, E)\) has \(E(G)\) as the vertex set, where two vertices are adjacent if they are incident edges in \(G\). We extend our results to trees as trees and its line graphs are block graphs. We prove that if a tree is a lower bound block graph then, under certain conditions, its line graph is also a lower bound block graph, and vice versa. Consequently, we show that the line graphs of many known lower bound trees, excluding paths, are lower bound block graphs.Antipodal number of full \(m\)-ary treeshttps://zbmath.org/1517.051542023-09-22T14:21:46.120933Z"Basunia, Alamgir Rahaman"https://zbmath.org/authors/?q=ai:basunia.alamgir-rahaman"Das, Satyabrata"https://zbmath.org/authors/?q=ai:das.satyabrata"Saha, Laxman"https://zbmath.org/authors/?q=ai:saha.laxman"Tiwary, Kalishankar"https://zbmath.org/authors/?q=ai:tiwary.kalishankarSummary: Let \(G = (V(G), E(G))\) be a simple connected graph. For any two distinct vertices \(u\) and \(v\), \(d(u, v)\) represents the distance between \(u\) and \(v\). Suppose \(k\) be any positive integer with \(1 \leqslant k \leqslant \operatorname{diam}(G)\), where the \(\operatorname{diam}(G)\) represents the diameter of \(G\). A radio \(k\)-labeling of \(G\) is a mapping \(f : V(G) \to \{0, 1, 2, \ldots \}\) such that \(| f(u) - f(v) | \geqslant k + 1 - d(u, v)\) for each pair of distinct vertices \(u\) and \(v\) of \(G\). The absolute difference of the largest and smallest values in \(f(V(G))\) is termed as the span of \(f\), and is denoted by \(\operatorname{span}(f)\). The antipodal number is the minimum span of a radio \(( \operatorname{diam} ( \operatorname{G} ) - 1)\)-labeling of \(G\) and the radio number is the minimum span of a radio \(\operatorname{diam}(G)\)-labeling of \(G\). In this article we determine the antipodal number of the \(m\)-ary tree for any \(m \geq 3\) with any height \(h \geq 3\) and construct explicitly an optimal antipodal labeling.On the role of 3s for the 1-2-3 conjecturehttps://zbmath.org/1517.051552023-09-22T14:21:46.120933Z"Bensmail, Julien"https://zbmath.org/authors/?q=ai:bensmail.julien"Fioravantes, Foivos"https://zbmath.org/authors/?q=ai:fioravantes.foivos"Mc Inerney, Fionn"https://zbmath.org/authors/?q=ai:mc-inerney.fionnSummary: The 1-2-3 Conjecture states that every connected graph different from \(K_2\) admits a proper 3-(edge-)labelling, i.e., can have its edges labelled with 1, 2, 3 so that no two adjacent vertices are incident to the same sum of labels. In connection with some recent optimisation variants of this conjecture, in this paper, we investigate the role of the label 3 in proper 3-labellings of graphs. An intuition from previous investigations is that, in general, it should always be possible to produce proper 3-labellings assigning label 3 to a only few edges.
We prove that, for every \(p \geq 0\), there are various graphs needing at least \(p\) 3s in their proper 3-labellings. Actually, deciding whether a given graph can be properly 3-labelled with \(p\) 3s is NP-complete for every \(p \geq 0\). We also focus on classes of 3-chromatic graphs. For various classes of such graphs (cacti, cubic graphs, triangle-free planar graphs, etc.), we prove that there is no \(p \geq 1\) such that all their graphs admit proper 3-labellings assigning label 3 to at most \(p\) edges. In such cases, we provide lower and upper bounds on the number of 3s needed.An inductive approach to strongly antimagic labelings of graphshttps://zbmath.org/1517.051562023-09-22T14:21:46.120933Z"Liu, Daphne Der-Fen"https://zbmath.org/authors/?q=ai:liu.daphne-der-fen"Lossada, Vicente"https://zbmath.org/authors/?q=ai:lossada.vicenteSummary: An antimagic labeling for a graph \(G\) with \(m\) edges is a bijection \(f:E(G)\to\{1,2,\dots,m\}\) so that \(\varphi_f(u)\ne\varphi_f(v)\) holds for any pair of distinct vertices \(u,v\in V(G)\), where \(\varphi_f(x)=\sum_{e\cap x\ne\emptyset}f(e)\). A strongly antimagic labeling is an antimagic labeling with an additional condition: For any \(u,v\in V(G)\), if \(\deg(u)>\deg(v)\), then \(\varphi_f(u)>\varphi_f(v)\). A graph \(G\) is strongly antimagic if it admits a strongly antimagic labeling. We present inductive properties of strongly antimagic labelings of graphs. This approach leads to simplified proofs that spiders and double spiders are strongly antimagic, previously shown by \textit{J.-L. Shang} [Ars Comb. 118, 367--372 (2015; Zbl 1349.05303)] and \textit{T.-Y. Huang} [Antimagic labeling on spiders. Taipei: Department of Mathematics, National Taiwan University (Master's Thesis) (2015)], and by \textit{F.-H. Chang} et al. [Indian J. Discrete Math. 6, No. 1, 43--68 (2020; Zbl 1471.05096)], respectively. We fix a subtle error in [Chang, loc. cit.]. Further, we prove certain level-wise regular trees, cycle spiders and cycle double spiders are all strongly antimagic.On magic distinct labellings of simple graphshttps://zbmath.org/1517.051572023-09-22T14:21:46.120933Z"Xin, Guoce"https://zbmath.org/authors/?q=ai:xin.guoce"Xu, Xinyu"https://zbmath.org/authors/?q=ai:xu.xinyu"Zhang, Chen"https://zbmath.org/authors/?q=ai:zhang.chen"Zhong, Yueming"https://zbmath.org/authors/?q=ai:zhong.yueming\textit{J. A. MacDougall} et al. [Util. Math. 61, 3--21 (2002; Zbl 1008.05135)] have introduced the concept of magic labelling of simple graphs. A magic labelling of a graph $G$ with magic sum $s$ is a labelling of the edges of $G$ by nonnegative integers such that for each vertex $v\in V$, the sum of labels of all edges incident to $v$ is equal to the same number $s$. A magic distinct labelling is a magic labelling whose labels are distinct. It is said to be pure if the labels are $1,2,\dots,n$. The authors consider here the complete construction of all magic labellings of a given graph $G$. They illustrate it in detail by dealing with three regular graphs and a non-regular graph. They give combinatorial proofs. The structure they find here can be used to enumerate the corresponding magic distinct labellings. The idea of using the generating function is highly laudable.
Reviewer: V. Yegnanarayanan (Chennai)Largest component of subcritical random graphs with given degree sequencehttps://zbmath.org/1517.051582023-09-22T14:21:46.120933Z"Coulson, Matthew"https://zbmath.org/authors/?q=ai:coulson.matthew"Perarnau, Guillem"https://zbmath.org/authors/?q=ai:perarnau.guillemSummary: We study the size of the largest component of two models of random graphs with prescribed degree sequence, the configuration model (CM) and the uniform model (UM), in the (barely) subcritical regime. For the CM, we give upper bounds that are asymptotically tight for certain degree sequences. These bounds hold under mild conditions on the sequence and improve previous results of \textit{H. Hatami} and \textit{M. Molloy} [Random Struct. Algorithms 41, No. 1, 99--123 (2012; Zbl 1247.05218)] on the barely subcritical regime. For the UM, we give weaker upper bounds that are tight up to logarithmic terms but require no assumptions on the degree sequence. In particular, the latter result applies to degree sequences with infinite variance in the subcritical regime.Many nodal domains in random regular graphshttps://zbmath.org/1517.051592023-09-22T14:21:46.120933Z"Ganguly, Shirshendu"https://zbmath.org/authors/?q=ai:ganguly.shirshendu"McKenzie, Theo"https://zbmath.org/authors/?q=ai:mckenzie.theo"Mohanty, Sidhanth"https://zbmath.org/authors/?q=ai:mohanty.sidhanth"Srivastava, Nikhil"https://zbmath.org/authors/?q=ai:srivastava.nikhilSummary: Let \(G\) be a random \(d\)-regular graph on \(n\) vertices. We prove that for every constant \(\alpha > 0\), with high probability every eigenvector of the adjacency matrix of \(G\) with eigenvalue less than \(-2\sqrt{d-2}-\alpha\) has \(\Omega (n/\text{polylog}(n))\) nodal domains.On the minimum bisection of random 3-regular graphshttps://zbmath.org/1517.051602023-09-22T14:21:46.120933Z"Lichev, Lyuben"https://zbmath.org/authors/?q=ai:lichev.lyuben"Mitsche, Dieter"https://zbmath.org/authors/?q=ai:mitsche.dieterSummary: In this paper we give new bounds on the bisection width of random 3-regular graphs on \(n\) vertices. The main contribution is a new lower bound of \(0.103295n\) based on a first moment method together with a structural analysis of the graph, thereby improving a 27-year-old result of \textit{A. V. Kostochka} and \textit{L. S. Melnikov} [in: Вероятностные методы дискретной\ математики. Труды третьей\ Петрозаводской\ конференции. Петрозаводск (Россия), 11--16 мая 1992 г. Moskva: TVP; Utrecht: VSP. 251--265 (1993; Zbl 0811.05035)]. We also give a complementary upper bound of \(0.139822n\) by combining a result of Lyons with original combinatorial insights. Developping this approach further, we obtain a non-rigorous improved upper bound with the help of Monte Carlo simulations.Hitting times for random walks on tricyclic graphshttps://zbmath.org/1517.051612023-09-22T14:21:46.120933Z"Zhu, Xiao-Min"https://zbmath.org/authors/?q=ai:zhu.xiaomin"Yang, Xu"https://zbmath.org/authors/?q=ai:yang.xuSummary: Let \(G\) be a simple connected graph and \(x\), \(y \in V(G)\). Let \(H_G (x,y)\) be the expected hitting time from \(x\) to \(y\) in \(G\) and \(\varphi(G)\) be the hitting time of \(G\), where \(\varphi(G) = \max\{H_G(x,y) : x,y \in V(G)\}\) A tricyclic graph is a simple connected graph that the edge number equals the vertex number plus two. Let \(\mathcal{T}(n)\) be the set of all \(n\)-vertex tricyclic graphs. In this article, we will determine the extremal graphs for hitting times among all \(n\)-vertex tricyclic graphs. Moreover, if \(G \in \mathcal{T}(n)\) then we will obtain sharp upper and lower bounds for \(\varphi(G)\).Degree centrality and root finding in growing random networkshttps://zbmath.org/1517.051622023-09-22T14:21:46.120933Z"Banerjee, Sayan"https://zbmath.org/authors/?q=ai:banerjee.sayan"Huang, Xiangying"https://zbmath.org/authors/?q=ai:huang.xiangyingSummary: We consider growing random networks \({\{{\mathcal{G}_n}\}_{n\ge 1}}\) where, at each time, a new vertex attaches itself to a collection of existing vertices via a fixed number \(m\ge 1\) of edges, with probability proportional to a function \(f\) (called attachment function) of their degree. It was shown in [\textit{S. Banerjee} and \textit{S. Bhamidi}, Probab. Theory Relat. Fields 180, No. 3--4, 891--953 (2021; Zbl 1491.60015)] that such network models exhibit two distinct regimes: (i) the persistent regime, corresponding to \(\sum_{i=1}^\infty f(i)^{-2} < \infty\), where the top \(K\) maximal degree vertices fixate over time for any given \(K\), and (ii) the non-persistent regime, with \(\sum_{i=1}^\infty f(i)^{-2} = \infty\), where the identities of these vertices keep changing infinitely often over time. In this article, we develop root finding algorithms using the empirical degree structure and local network information based on a snapshot of such a network at some large time. In the persistent regime, the algorithm is purely based on degree centrality, that is, for a given error tolerance \(\varepsilon \in (0,1)\), there exists \(K_{\varepsilon } \in \mathbb{N}\) such that for any \(n\ge 1\), the confidence set for the root in \(\mathcal{G}_n \), which contains the root with probability at least \(1-\varepsilon \), consists of the top \(K_{\varepsilon}\) maximal degree vertices. In particular, the size of the confidence set is stable in the network size. Upper and lower bounds on \(K_{\varepsilon }\) are explicitly characterized in terms of the error tolerance \(\varepsilon\) and the attachment function \(f\). In the non-persistent regime, for an appropriate choice of \({r_n}\to \infty\) at a rate much smaller than the diameter of the network, the neighborhood of radius \({r_n}\) around the maximal degree vertex is shown to contain the root with high probability, and a size estimate for this set is obtained. It is shown that, when \(f(k)=k^{\alpha}\), \(k\ge 1\), for any \(\alpha \in (0,1/ 2]\), this size grows at a smaller rate than any positive power of the network size.Average Fermat distance of a pseudo-fractal hierarchical scale-free networkhttps://zbmath.org/1517.051632023-09-22T14:21:46.120933Z"Peng, Lulu"https://zbmath.org/authors/?q=ai:peng.lulu"Zeng, Cheng"https://zbmath.org/authors/?q=ai:zeng.cheng"Chen, Dirong"https://zbmath.org/authors/?q=ai:chen.dirong"Xue, Yumei"https://zbmath.org/authors/?q=ai:xue.yumei"Zhao, Zixuan"https://zbmath.org/authors/?q=ai:zhao.zixuan(no abstract)Maximum weight independent sets for (\(S_{1,2,4}\), triangle)-free graphs in polynomial timehttps://zbmath.org/1517.051642023-09-22T14:21:46.120933Z"Brandstädt, Andreas"https://zbmath.org/authors/?q=ai:brandstadt.andreas"Mosca, Raffaele"https://zbmath.org/authors/?q=ai:mosca.raffaeleSummary: The Maximum Weight Independent Set (MWIS) problem on finite undirected graphs with vertex weights asks for a set of pairwise nonadjacent vertices of maximum weight sum. MWIS is one of the most investigated and most important algorithmic graph problems; it is well known to be NP-complete, and it remains NP-complete even under various strong restrictions such as for triangle-free graphs. Its complexity for \(P_k\)-free graphs, \(k \geq 7\), is an open problem. In [\textit{A. Brandstädt} and \textit{R. Mosca}, Discrete Appl. Math. 236, 57--65 (2018; Zbl 1377.05185)], it is shown that MWIS can be solved in polynomial time for (\( P_7\),triangle)-free graphs. This result is extended by \textit{F. Maffray} and \textit{L. Pastor} [Discrete Math. 341, No. 5, 1449--1458 (2018; Zbl 1383.05145)] showing that MWIS can be solved in polynomial time for (\( P_7\),bull)-free graphs. In the same paper, they also showed that MWIS can be solved in polynomial time for (\( S_{1 , 2 , 3}\),bull)-free graphs. In this paper, using a similar approach as in [Brandstädt and Mosca, loc. cit.], we show that MWIS can be solved in polynomial time for (\( S_{1 , 2 , 4}\),triangle)-free graphs which generalizes the result for (\(P_7\),triangle)-free graphs.The Weisfeiler-Leman algorithm and recognition of graph propertieshttps://zbmath.org/1517.051652023-09-22T14:21:46.120933Z"Fuhlbrück, Frank"https://zbmath.org/authors/?q=ai:fuhlbruck.frank"Köbler, Johannes"https://zbmath.org/authors/?q=ai:kobler.johannes"Ponomarenko, Ilia"https://zbmath.org/authors/?q=ai:ponomarenko.ilya"Verbitsky, Oleg"https://zbmath.org/authors/?q=ai:verbitsky.olegSummary: The \(k\)-dimensional Weisfeiler-Leman algorithm (\(k\)-WL) is a very useful combinatorial tool in graph isomorphism testing. We address the applicability of \(k\)-WL to recognition of graph properties. Let \(G\) be an input graph with \(n\) vertices. We show that, if \(n\) is prime, then vertex-transitivity of \(G\) can be seen in a straightforward way from the output of 2-WL on \(G\) and on the vertex-individualized copies of \(G\). This is perhaps the first non-trivial example of using the Weisfeiler-Leman algorithm for recognition of a natural graph property rather than for isomorphism testing. On the other hand, we show that, if \(n\) is divisible by 16, then \(k\)-WL is unable to distinguish between vertex-transitive and non-vertex-transitive graphs with \(n\) vertices unless \(k = \Omega(\sqrt{ n})\). Similar results are obtained for recognition of arc-transitivity. Our lower bounds are based on an analysis of the Cai-Fürer-Immerman construction, which might be of independent interest. In particular, we provide sufficient conditions under which the Cai-Fürer-Immerman graphs can be made colorless.The Weisfeiler-Leman algorithm and recognition of graph propertieshttps://zbmath.org/1517.051662023-09-22T14:21:46.120933Z"Fuhlbrück, Frank"https://zbmath.org/authors/?q=ai:fuhlbruck.frank"Köbler, Johannes"https://zbmath.org/authors/?q=ai:kobler.johannes"Ponomarenko, Ilia"https://zbmath.org/authors/?q=ai:ponomarenko.ilya"Verbitsky, Oleg"https://zbmath.org/authors/?q=ai:verbitsky.olegSummary: The \(k\)-dimensional Weisfeiler-Leman algorithm (\(k\)-WL) is a very useful combinatorial tool in graph isomorphism testing. We address the applicability of \(k\)-WL to recognition of graph properties. Let \(G\) be an input graph with \(n\) vertices. We show that, if \(n\) is prime, then vertex-transitivity of \(G\) can be seen in a straightforward way from the output of 2-WL on \(G\) and on the vertex-individualized copies of \(G\). This is perhaps the first non-trivial example of using the Weisfeiler-Leman algorithm for recognition of a natural graph property rather than for isomorphism testing. On the other hand, we show that, if \(n\) is divisible by 16, then \(k\)-WL is unable to distinguish between vertex-transitive and non-vertex-transitive graphs with \(n\) vertices unless \(k=\varOmega (\sqrt{n})\).
For the entire collection see [Zbl 1507.68025].A linear time algorithm for finding maximal planar subgraphshttps://zbmath.org/1517.051672023-09-22T14:21:46.120933Z"Hsu, Wen-Lian"https://zbmath.org/authors/?q=ai:hsu.wen-lian|hsu.wenlianThe author proposes a new algorithm for finding a maximal planar subgraph of a given graph. There exist several algorithms in the literature for both planarity testing and finding the maximal planar subgraph. The chief claim of the paper is the simplicity of their approach. Additionally, the proposed algorithm is asymptotically optimal.
The problem of finding a maximal planar subgraph is extremely important since it arises in a number of domains such as graph drawing, VLSI design, and so on. Efficient algorithms are therefore of paramount importance. At the same time, a theoretically efficient algorithm that uses complex data structures may not be useful in a practical setting.
The paper provides a detailed description of the author's planarity testing algorithm. This greatly enhances the readability of the paper. The exposition is straightforward and illustrative.
The algorithms themselves are somewhat complex, although based on fundamental algorithms such as depth-first search and vertex addition.
On the whole, this paper is a valuable addition to the literature on the subject.
For the entire collection see [Zbl 0856.00037].
Reviewer: K. Subramani (Morgantown)Algorithms for the rainbow vertex coloring problem on graph classeshttps://zbmath.org/1517.051682023-09-22T14:21:46.120933Z"Lima, Paloma T."https://zbmath.org/authors/?q=ai:lima.paloma-t"van Leeuwen, Erik Jan"https://zbmath.org/authors/?q=ai:van-leeuwen.erik-jan"van der Wegen, Marieke"https://zbmath.org/authors/?q=ai:van-der-wegen.mariekeSummary: Given a vertex-colored graph, we say a path is a rainbow vertex path if all its internal vertices have distinct colors. The graph is rainbow vertex-connected if there is a rainbow vertex path between every pair of its vertices. In the Rainbow Vertex Coloring (RVC) problem we want to decide whether the vertices of a given graph can be colored with at most \(k\) colors so that the graph becomes rainbow vertex-connected. This problem is known to be NP-complete even in very restricted scenarios, and very few efficient algorithms are known for it. In this work, we give polynomial-time algorithms for RVC on permutation graphs, powers of trees and split strongly chordal graphs. The algorithm for the latter class also works for the strong variant of the problem, where the rainbow vertex paths between each vertex pair must be shortest paths. We complement the polynomial-time solvability results for split strongly chordal graphs by showing that, for any fixed \(p\geq 3\) both variants of the problem become NP-complete when restricted to split \((S_3,\dots,S_p)\)-free graphs, where \(S_q\) denotes the \(q\)-sun graph.
For the entire collection see [Zbl 1445.68013].On the complexity of the planar edge-disjoint paths problem with terminals on the outer boundaryhttps://zbmath.org/1517.051692023-09-22T14:21:46.120933Z"Schwärzler, Werner"https://zbmath.org/authors/?q=ai:schwarzler.wernerSummary: It is shown that both the undirected and the directed edge-disjoint paths problem are NP-complete, if the supply graph is planar and all edges of the demand graph are incident with vertices lying on the outer boundary of the supply graph. In the directed case, the problem remains NP-complete, if in addition the supply graph is acyclic. The undirected case solves open problem no. 56 of \textit{A. Schrijver}'s book [Combinatorial optimization. Polyhedra and efficiency (3 volumes). Berlin: Springer (2003; Zbl 1041.90001)].A certifying and dynamic algorithm for the recognition of proper circular-arc graphshttps://zbmath.org/1517.051702023-09-22T14:21:46.120933Z"Soulignac, Francisco J."https://zbmath.org/authors/?q=ai:soulignac.francisco-juanSummary: We present a dynamic algorithm for the recognition of proper circular-arc (PCA) graphs, that supports the insertion and removal of vertices (together with its incident edges). The main feature of the algorithm is that it outputs a minimally non-PCA induced subgraph when the insertion of a vertex fails. Each operation cost \(O(\log n+d)\) time, where \(n\) is the number vertices and \(d\) is the degree of the modified vertex. When removals are disallowed, each insertion is processed in \(O(d)\) time. The algorithm also provides two constant-time operations to query if the dynamic graph is proper Helly (PHCA) or proper interval (PIG). When the dynamic graph is not PHCA (resp. PIG), a minimally non-PHCA (resp. non-PIG) induced subgraph is obtained.Characterizing 3-sets in union-closed familieshttps://zbmath.org/1517.051712023-09-22T14:21:46.120933Z"Pulaj, Jonad"https://zbmath.org/authors/?q=ai:pulaj.jonadSummary: Let \([n] := \{1, 2, \dots, n\}\) and let a \(k\)-set denote a set of cardinality \(k\). A family of sets is union-closed (UC) if the union of any two sets in the family is also in the family. Frankl's conjecture states that for any nonempty UC family \(\mathcal{F} \subseteq 2^{[n]}\) such that \(\mathcal{F} \neq \{\emptyset\}\), there exists an element \(i \in [n]\) that is contained in at least half the sets of \(\mathcal{F}\), where \(2^{[n]}\) denotes the power set on \([n]\). The 3-sets conjecture of Morris states that the smallest number of distinct 3-sets (whose union is an \(n\)-set) that ensure Frankl's conjecture is satisfied (in an element of the \(n\)-set) for any UC family that contains them is \(\lfloor n/2 \rfloor + 1\) for all \(n \geq 4\). For an UC family \(\mathcal{A} \subseteq 2^{[n]}\), Poonen's theorem characterizes the existence of weights on \([n]\) which ensure all UC families that contain \(\mathcal{A}\) satisfy Frankl's conjecture, however the determination of such weights for specific \(\mathcal{A}\) is nontrivial even for small \(n\). We classify families of 3-sets on \(n \leq 9\) using a polyhedral interpretation of Poonen's theorem and exact rational integer programming. This yields a proof of the 3-sets conjecture.A Kruskal-Katona-type theorem for graphs: \(q\)-Kneser graphshttps://zbmath.org/1517.051722023-09-22T14:21:46.120933Z"Wang, Jun"https://zbmath.org/authors/?q=ai:wang.jun"Xu, Ao"https://zbmath.org/authors/?q=ai:xu.ao"Zhang, Huajun"https://zbmath.org/authors/?q=ai:zhang.huajunSummary: The ``Kruskal-Katona-type problem for a graph \(G\)'' concerned here is to describe subsets of vertices of \(G\) that have minimum number of neighborhoods with respect to their sizes. In this paper, we establish a Kruskal-Katona-type theorem for the \(q\)-Kneser graphs, whose vertex set consists of all \(k\)-dimensional subspaces of an \(n\)-dimensional linear space over a \(q\)-element field, two subspaces are adjacent if they have the trivial intersection. It includes as a special case the Erdős-Ko-Rado theorem for intersecting families in finite vector spaces and yields a short proof of the Hilton-Milner theorem for nontrivial intersecting families in finite vector spaces.On the rainbow numbers of \(\mathbb{Z}_n\) for \(x_1 + x_2 = 4x_3\)https://zbmath.org/1517.051732023-09-22T14:21:46.120933Z"El Turkey, Houssein"https://zbmath.org/authors/?q=ai:el-turkey.houssein"Waskiewicz, Nathan"https://zbmath.org/authors/?q=ai:waskiewicz.nathanSummary: An exact \(r\)-coloring of a set \(S\) is a surjective function \(c:S\to\{1, 2, \dots, r\}\). Given an equation \(eq\), a solution in \(S\) is a rainbow solution if each element is colored distinctly by the coloring \(c\). The rainbow number of a set \(S\) for equation \(eq\) is the smallest integer \(r\) such that every exact \(r\)-coloring of \(S\) contains a rainbow solution to \(eq\). The rainbow numbers of \(\mathbb{Z}_p\), for prime \(p\), for the equation \(x_1 + x_2 = 4x_3\) are known to be either 3 or 4. This paper investigates which primes yield rainbow number 3 or 4. Additionally, the rainbow numbers of \(\mathbb{Z}_n\) for this equation are discussed.Some new results on monochromatic sums and products in the rationalshttps://zbmath.org/1517.051742023-09-22T14:21:46.120933Z"Hindman, Neil"https://zbmath.org/authors/?q=ai:hindman.neil"Ivan, Maria-Romina"https://zbmath.org/authors/?q=ai:ivan.maria-romina"Leader, Imre"https://zbmath.org/authors/?q=ai:leader.imreThe main result of the paper states that there exists a finite colouring of the rational numbers with the following property: There is no infinite set of rationals such that the set of its finite sums and products is monochromatic and the set of primes that divide the denominators of its terms is finite. In other words, given an infinite set of rational numbers whose denominators are divisible only by a finite number of primes, then the set of its finite sums and products it not monochromatic.
Besides this amazing result, they prove several auxiliary results concerning colorings of the natural numbers and the real numbers. For example, they prove that there exists a finite colouring of the natural numbers, such that there is no injective sequence \((x_n)_{n\geq 1}\) of natural numbers with the property that for any \(1\leq n<m\), all numbers \(x_n+x_m\) and \(x_n\cdot x_m\) have the same colour.
Reviewer: Lorenz Halbeisen (Zürich)The Tuza-Vestergaard theoremhttps://zbmath.org/1517.051752023-09-22T14:21:46.120933Z"Henning, Michael A."https://zbmath.org/authors/?q=ai:henning.michael-anthony"Löwenstein, Christian"https://zbmath.org/authors/?q=ai:lowenstein.christian"Yeo, Anders"https://zbmath.org/authors/?q=ai:yeo.andersSummary: The transversal number \(\tau(H)\) of a hypergraph \(H\) is the minimum number of vertices that intersect every edge of \(H\). A 6-uniform hypergraph has all edges of size 6. On 10 November 2000, \textit{Z. Tuza} and \textit{P. D. Vestergaard} [Discuss. Math., Graph Theory 22, No. 1, 199--210 (2002; Zbl 1016.05057)] conjectured that if \(H\) is a 3-regular 6-uniform hypergraph of order \(n\), then \(\tau(H)\leq\frac{1}{4}n\). In this paper we prove this conjecture, which has become known as the Tuza-Vestergaard conjecture.On connectivity of conditional configuration graphs under destructionhttps://zbmath.org/1517.051762023-09-22T14:21:46.120933Z"Leri, M. M."https://zbmath.org/authors/?q=ai:leri.marina-mThe configuration model for random graphs is the most robust model that permits the study of a random model for graphs with a given degree pattern. These are sometimes more relevant in contrast to the ubiquitous Erdős-Renyi model as many networks in real life (social networks, telecommunication networks, etc) are better modeled by `power-law' graphs. More precisely, suppose \(\tau>1\) is a real parameter, and consider the random variable \(\xi\) whose distribution is given by \(\mathbb{P}_{\tau}(\xi=k)=k^{-\tau}-(k+1)^{-\tau}\), for \(k\in\mathbb{N}\). Now consider the configuration model (of a multigraph with loops) with \(N\) vertices and degrees \(\xi_1,\ldots,\xi_N\) where \(\xi\) are i.i.d to the distribution \(\mathbb{P}_{\tau}\) described above. One of the main interests in these models is to see when these graphs are connected, and if so, how many `knocks' they can take before losing connectivity. In other words, suppose we sequentially delete vertices from a \(\mathbb{P}_{\tau}\)-random graph. The question of interest is: How many steps of deletions can we make before we end up with a graph that is no longer connected? The vertices to be deleted are again chosen randomly, and through one of two means; pick the largest degree vertex, or pick a random vertex.
The current paper investigates these questions through some computer simulations, with a regression model. The author shows the results of these simulations for varying values of \(\tau\) and shows regression models for the probability of disconnecting the graph after deleting \(r\%\) of the vertices, under both these models of vertex selection for deletion.
Reviewer: Niranjan Balachandran (Mumbai)Free fermions and Schur expansions of multi-Schur functionshttps://zbmath.org/1517.051772023-09-22T14:21:46.120933Z"Iwao, Shinsuke"https://zbmath.org/authors/?q=ai:iwao.shinsukeSummary: Multi-Schur functions are symmetric functions that generalize the supersymmetric Schur functions, the flagged Schur functions, and the refined dual Grothendieck functions, which have been intensively studied by \textit{A. Lascoux} [Symmetric functions and combinatorial operators on polynomials. Providence, RI: American Mathematical Society (AMS) (2003; Zbl 1039.05066)]. In this paper, we give a new free-fermionic presentation of them. The multi-Schur functions are indexed by a partition and two ``tuples of tuples'' of indeterminates. We construct a family of linear bases of the fermionic Fock space that are indexed by such data and prove that they correspond to the multi-Schur functions through the boson-fermion correspondence. By focusing on some special bases, which we call refined bases, we give a straightforward method of expanding a multi-Schur function in the refined dual Grothendieck polynomials. We also present a sufficient condition for a multi-Schur function to have its Hall-dual function in the completed ring of symmetric functions.Row-strict dual immaculate functionshttps://zbmath.org/1517.051782023-09-22T14:21:46.120933Z"Niese, Elizabeth"https://zbmath.org/authors/?q=ai:niese.elizabeth-m"Sundaram, Sheila"https://zbmath.org/authors/?q=ai:sundaram.sheila"van Willigenburg, Stephanie"https://zbmath.org/authors/?q=ai:van-willigenburg.stephanie-j"Vega, Julianne"https://zbmath.org/authors/?q=ai:vega.julianne"Wang, Shiyun"https://zbmath.org/authors/?q=ai:wang.shiyunSummary: We define a new basis of quasisymmetric functions, the row-strict dual immaculate functions, as the generating function of a particular set of tableaux. We show that this definition gives a function that can also be obtained by applying the involution \(\psi\) to the dual immaculate functions of \textit{C. Berg} et al. [Can. J. Math. 66, No. 3, 525--565 (2014; Zbl 1291.05206)], and we establish numerous combinatorial properties for our functions. We give an equivalent formulation of our functions via Bernstein-like operators, in a similar fashion to [loc. cit.]. We conclude the paper by defining skew dual immaculate functions and hook dual immaculate functions, and establishing combinatorial properties for them.Cyclic descents, matchings and Schur-positivityhttps://zbmath.org/1517.051792023-09-22T14:21:46.120933Z"Adin, Ron M."https://zbmath.org/authors/?q=ai:adin.ron-m"Roichman, Yuval"https://zbmath.org/authors/?q=ai:roichman.yuvalSummary: A new descent set statistic on involutions, defined geometrically via their interpretation as matchings, is introduced in this paper, and shown to be equidistributed with the standard one. This concept is then applied to construct explicit cyclic descent extensions on involutions, standard Young tableaux and Motzkin paths. Schur-positivity of the associated quasisymmetric functions follows.Permutations whose reverse shares the same recording tableau in the RS correspondencehttps://zbmath.org/1517.051802023-09-22T14:21:46.120933Z"Ervin, Tucker J."https://zbmath.org/authors/?q=ai:ervin.tucker-j"Jackson, Blake"https://zbmath.org/authors/?q=ai:jackson.blake"Lane, Jay"https://zbmath.org/authors/?q=ai:lane.jay"Lee, Kyungyong"https://zbmath.org/authors/?q=ai:lee.kyungyong"Nguyen, Son Dang"https://zbmath.org/authors/?q=ai:nguyen.son-dang"O'Donohue, Jack"https://zbmath.org/authors/?q=ai:odonohue.jack"Vaughan, Michael"https://zbmath.org/authors/?q=ai:vaughan.michaelSummary: The RS correspondence is a bijection between permutations and pairs of standard Young tableaux with identical shape, where the tableaux are commonly denoted \(P\) (insertion) and \(Q\) (recording). It has been an open problem to demonstrate
\[
|\{w \in \mathfrak{S}_n \mid Q(w) = Q(w^r)\}| = \begin{cases} 2^{\frac{n-1}{2}\binom{n-1}{\frac{n-1}{2}}}, & n \text{ odd,} \\ 0, & n \text{ even,}\end {cases}
\]
where \(w^r\) is the reverse permutation of \(w\). First we show that for each \(w\) where \(Q(w) = Q(w^r)\) the recording tableau \(Q(w)\) has a symmetric hook shape and satisfies a certain simple property. From these two results, we succeed in proving the desired identity.Derivations in group algebras and combinatorial invariants of groupshttps://zbmath.org/1517.051812023-09-22T14:21:46.120933Z"Arutyunov, Andronick"https://zbmath.org/authors/?q=ai:arutyunov.andronick-aramovichSummary: This article is devoted to the study of derivations in group algebras. Ideals of inner and quasi-inner derivations are constructed, which makes it possible to study the algebras of outer and quasi-outer derivations. We establish a relationship between derivations and characters on the groupoid of the adjoint action. Moreover, we study a connection between the structure of algebras and the number of ends of the conjugacy diagram and, as a consequence, the number of ends of the original group. The results obtained make it possible in the final analysis to give estimates for the dimension of the algebras of outer and quasi-outer derivations using only the combinatorial properties of the group. It is also possible to obtain information about the Hochschild one-cohomology.A note on the action of the Hecke group \(H(2)\) on subsets of the form \(\mathbb{Q}^{\ast}(\sqrt{n})\)https://zbmath.org/1517.051822023-09-22T14:21:46.120933Z"Cimpoeaş, Mircea"https://zbmath.org/authors/?q=ai:cimpoeas.mirceaSummary: Here, we study the action of the groups \(H(\lambda)\) generated by the linear fractional transformations \(x:z\mapsto -\frac{1}{z}\) and \(w:z\mapsto z+\lambda\) and \(\lambda\) is a positive integer, on the subsets \(\mathbb{Q}^{\ast}(\sqrt{n})=\{\frac{a+\sqrt{n}}{c}\mid a,b=\frac{a^2 -n}{c},c\in\mathbb{Z}\}\cup\{0,1,\infty\}\) and \(|n|\) is a square-free positive integer. We prove that this action has a finite number of orbits if and only if \(\lambda =1\) or \(\lambda =2\). Moreover, we give an upper bound for the number of orbits for \(\lambda =2\).Regular Cayley maps of elementary abelian \(p\)-groups: classification and enumerationhttps://zbmath.org/1517.051832023-09-22T14:21:46.120933Z"Du, Shaofei"https://zbmath.org/authors/?q=ai:du.shaofei"Yu, Hao"https://zbmath.org/authors/?q=ai:yu.hao.3|yu.hao.1|yu.hao.4|yu.hao.2|yu.hao"Luo, Wenjuan"https://zbmath.org/authors/?q=ai:luo.wenjuanSummary: Recently, regular Cayley maps of cyclic groups and dihedral groups have been classified in [\textit{M. D. E. Conder} and \textit{T. W. Tucker}, Trans. Am. Math. Soc. 366, No. 7, 3585--3609 (2014; Zbl 1290.05160)] and [\textit{I. Kovács} and \textit{Y. S. Kwon}, J. Comb. Theory, Ser. B 148, 84--124 (2021; Zbl 1459.05119)], respectively. A natural question is to classify regular Cayley maps of elementary abelian \(p\)-groups \(\mathbb{Z}_p^n\). In this paper, a complete classification of regular Cayley maps of \(\mathbb{Z}_p^n\) is given and moreover, the number of these maps and their genera are enumerated.Combinatorial properties for a class of simplicial complexes extended from pseudo-fractal scale-free webhttps://zbmath.org/1517.051842023-09-22T14:21:46.120933Z"Xie, Zixuan"https://zbmath.org/authors/?q=ai:xie.zixuan"Wang, Yucheng"https://zbmath.org/authors/?q=ai:wang.yucheng"Xu, Wanyue"https://zbmath.org/authors/?q=ai:xu.wanyue"Zhu, Liwang"https://zbmath.org/authors/?q=ai:zhu.liwang"Li, Wei"https://zbmath.org/authors/?q=ai:li.wei.258"Zhang, Zhongzhi"https://zbmath.org/authors/?q=ai:zhang.zhongzhi(no abstract)Quadrinomial-like versions for Wolstenholme, Morley and Glaisher congruenceshttps://zbmath.org/1517.110012023-09-22T14:21:46.120933Z"Belbachir, Hacène"https://zbmath.org/authors/?q=ai:belbachir.hacene"Otmani, Yassine"https://zbmath.org/authors/?q=ai:otmani.yassineMotivated by the famous congruence by \textit {J. Wolstenholme} (1862), valid for any odd prime \(p \geq 5\),
\[
\binom{2p-1}{p-1}\equiv 1 \pmod {p^3},
\]
and thanks to two further well-known congruences from, respectively, \textit {E. Lehmer} [Ann. Math. (2) 39, 350--360 (1938; Zbl 0019.00505)], and \textit {W. Ljunggren} et al. [11. Skand. Mat.-Kongr., Trondheim 1949, 42--54 (1952; Zbl 0048.27204)], the authors prove, via Fermat's Little Theorem and Taylor expansion, that
\[
\kappa_{p} \equiv 4+8 p^{2} \left(\frac{-1}{p} \right) E_{p-3} \pmod {p^{3}},
\]
where \(E_{n}\) is the \(n\)-th Euler number and \(\kappa_{p}\) is the central quadrinomial coefficient, i.e., the coefficient of \(x^{3p}\) in the polynomial expansion of \((1+x+x^2+x^3)^{2p}\).
Then, after employing three congruences by \textit {Z.-H. Sun} [Fibonacci Q. 40, No. 4, 345--351 (2002; Zbl 1009.11004)], a theorem by \textit {R. Tauraso} [J. Integer Seq. 19, No. 5, Article 16.5.4 (2016; Zbl 1417.11002)], a note by \textit {F. Morley} [Ann. Math. 9, 168--170 (1895; JFM 26.0208.02)], and a lemma by \textit {S. Mattarei} and \textit {R. Tauraso} [J. Number Theory 133, No. 1, 131--157 (2013; Zbl 1300.11020)], the authors establish that for any odd prime \(p \geq 5\),
\[
\kappa_{(p-1) / 2} \equiv \left( \frac{-2}{p} \right)+p \left(q_{p}(2) \left(\frac{7}{2} \left(\frac{-2}{p} \right)-3 \left( \frac{-1}{p} \right) \right)-2 \left(\frac{2}{p} \right) A_{p} \right) \pmod {p^{2}},
\]
where
\[
q_p(x) := \frac{x^{p-1}-1}{p}
\]
is the Fermat quotient, with \(x\) relatively prime to \(p\), and
\[
A_{p} := \frac{(-1)^{(p-1) / 2} P_{p}-(-8)^{(p-1) / 2}}{p},
\]
\( \left( P_n \right)_n \) being the Pell sequence.
Related to \(A_{p}\) is also a congruence provided by \textit {Kh. Hessami Pilehrood} et al. [Int. J. Number Theory 8, No. 7, 1789--1811 (2012; Zbl 1261.11001)] and here used to show that, for any positive integer \(n\) and any odd prime \(p \geq 5\),
\[
\left( \begin{array}{c} np-1 \\
p-1 \end{array} \right)_{3} \equiv \frac{1}{2} \left( \left( \frac{-1}{p} \right)+1 \right)+p q_{p}(2) \frac{n}{4} \left(5 \left( \frac{-1}{p} \right)+3 \right) \pmod {p^{2}},
\]
whose explicit source of inspiration is a result given by \textit {J. W. L. Glaisher} [Q. J. Math. 31, 1--35 (1900; JFM 30.0180.01)].
Reviewer: Enzo Bonacci (Latina)Supercongruences involving products of three binomial coefficientshttps://zbmath.org/1517.110042023-09-22T14:21:46.120933Z"Sun, Zhi-Hong"https://zbmath.org/authors/?q=ai:sun.zhihongSummary: Let \(p > 3\) be a prime, and let \(a\) be a rational \(p\)-adic integer. Using the WZ method we establish the congruences for \(\sum_{k=0}^{p-1}\binom{a}{k}\binom{-1-a}{k}\binom{2k}{k}\frac{w(k)}{4^k}\) modulo \(p^3\), where \(w(k)\in\{1, \frac{1}{k+1}, \frac{1}{(k+1)^2}, \frac{1}{2k-1}\}\). Taking \(a = -\frac{1}{2}, -\frac{1}{3}, -\frac{1}{4}, -\frac{1}{6}\) in the congruences confirms some conjectures posed by the author earlier.Triforce and cornershttps://zbmath.org/1517.110062023-09-22T14:21:46.120933Z"Fox, Jacob"https://zbmath.org/authors/?q=ai:fox.jacob"Sah, Ashwin"https://zbmath.org/authors/?q=ai:sah.ashwin"Sawhney, Mehtaab"https://zbmath.org/authors/?q=ai:sawhney.mehtaab-s"Stoner, David"https://zbmath.org/authors/?q=ai:stoner.david"Zhao, Yufei"https://zbmath.org/authors/?q=ai:zhao.yufeiThe \textit{triforce} is the \(3\)-graph (\(3\)-uniform hypergraph) with hyperedges \(123'\), \(12'3\) and \(1'23\). Thus the triforce is a \(6\)-vertex \(3\)-graph with three hyperedges. Let \(g(\delta)\) be the minimum density of triforces in a \(3\)-graph with hyperedge density \(\delta\). The first main result in the paper under review is that \(g(\delta)=\delta^{4-o(1)}\) but \(g(\delta)/\delta^4\to\infty\) as \(\delta\to0\). Via results of \textit{M. Mandache} [Math. Proc. Camb. Philos. Soc. 171, No. 3, 607--621 (2021; Zbl 1486.11015)], these estimates on \(g(\delta)\) translate to estimates on the density of so called \textit{popular corners} in \(G\times G\) for abelian groups \(G\) (see [\textit{M. Mandache}, Math. Proc. Camb. Philos. Soc. 171, No. 3, 607--621 (2021; Zbl 1486.11015)]). The authors also note that their proof of their estimates for~\(g(\delta)\) generalize to prove analogous estimates for the minimum density of \textit{\(k\)-forces} in \(k\)-graphs, where the \(k\)-force is the \(k\)-graph on \(2k\) vertices and hyperedges \(1'2\dots k\), \(12'\dots k\), \(\dots\), \(12\dots k'\): if \(g_k(\delta)\) is the corresponding extremal density function, then \(g_k(\delta)=\delta^{k+1-o(1)}\) but
\(g_k(\delta)/\delta^{k+1}\to\infty\) as \(\delta\to\infty\). One might expect these bounds to translate to good, polynomial bounds for popular \((k-1)\)-dimensional corners, but the authors prove a result (Theorem 1.6) that dashes any such hopes: for some absolute constant \(c>0\), given any fixed \(0<\delta<1/2\), for every sufficiently large \(N\), there is \(A\subset[N]^3\) with \(|A|\geq\delta N^3\) such that for every non-zero integer \(d\), there are at most \(\delta^{c\log1/\delta}N^3\) triples \((x,y,z)\) with \((x,y,z)\), \((x+d,y,z)\), \((x,y+d,z)\), and \((x,y,z+d)\) all in \(A\). The construction proving this result is related to a construction of sets of integers lacking popular differences for \(5\)-APs (see [\textit{V. Bergelson} et al., Invent. Math. 160, No. 2, 261--303 (2005; Zbl 1087.28007)]), which is in fact generalized in this paper to all \(5\)-point patterns in \(\mathbb N\) (Theorem 1.7).
Reviewer: Yoshiharu Kohayakawa (São Paulo)Fence tiling derived identities involving the metallonacci numbers squared or cubedhttps://zbmath.org/1517.110082023-09-22T14:21:46.120933Z"Allen, Michael A."https://zbmath.org/authors/?q=ai:allen.michael-a"Edwards, Kenneth"https://zbmath.org/authors/?q=ai:edwards.kennethSummary: We refer to the generalized Fibonacci sequence \((M^{(c)}_n)_{n\ge 0}\), where \(M^{(c)}_{n+1}=cM^{(c)}_n+M^{(c)}_{n-1}\) for \(n>0\) with \(M^{(c)}_0=0\), \(M^{(c)}_1=1\), for \(c=1,2,\dots\) as the \(c\)-metallonacci numbers. We consider the tiling of an \(n\)-board (an \(n\times 1\) rectangular board) with \(c\) colours of \(1/p\times 1\) tiles (with the shorter sides always aligned horizontally) and \((1/p,1-1/p)\)-fence tiles for \(p\in\mathbb{Z}^+\). A \((w,g)\)-fence tile is composed of two \(w\times 1\) sub-tiles separated by a \(g\times 1\) gap. The number of such tilings equals \((M^{(c)}_{n+1})^p\) and we use this result for the cases \(p=2, 3\) to devise straightforward combinatorial proofs of identities relating the metallonacci numbers squared or cubed to other combinations of metallonacci numbers. Special cases include relations between the Pell numbers cubed and the even Fibonacci numbers. Most of the identities derived here appear to be new.Some combinatorial aspects of bi-periodic incomplete Horadam sequenceshttps://zbmath.org/1517.110092023-09-22T14:21:46.120933Z"Belkhir, Amine"https://zbmath.org/authors/?q=ai:belkhir.amine"Tan, Elif"https://zbmath.org/authors/?q=ai:tan.elif"Dağh, Mehmet"https://zbmath.org/authors/?q=ai:dagh.mehmetSummary: We have recently introduced the bi-periodic incomplete Horadam numbers as a generalization of incomplete Horadam numbers, and studied their properties. In this paper, we provide some combinatorial interpretations of bi-periodic incomplete Horadam numbers by using the weighted tilings approach. We also define bi-periodic hyper Horadam numbers and show that each bi-periodic hyper Horadam number can be written as the difference of a bi-periodic Horadam number and a bi-periodic incomplete Horadam number.\(\mathbb{Q}\)-bonacci words and numbershttps://zbmath.org/1517.110122023-09-22T14:21:46.120933Z"Kirgizov, Sergey"https://zbmath.org/authors/?q=ai:kirgizov.sergeySummary: We present a quite curious generalization of multi-step Fibonacci numbers. For any positive rational \(q\), we enumerate binary words of length \(n\) whose maximal factors of the form \(0^a1^b\) satisfy \(a=0\) or \(aq>b\). When \(q\) is an integer we rediscover classical multistep Fibonacci numbers: Fibonacci, Tribonacci, Tetranacci, etc. When \(q\) is not an integer, obtained recurrence relations are connected to certain restricted integer compositions. We also discuss Gray codes for these words, and a possibly novel generalization of the golden ratio.Some identities involving degenerate Stirling numbers associated with several degenerate polynomials and numbershttps://zbmath.org/1517.110202023-09-22T14:21:46.120933Z"Kim, T. K."https://zbmath.org/authors/?q=ai:kim.tong-kuk|kim.tae-keuk|kim.tae-kyu|kim.timur-k|kim.tian-khoon|kim.tae-kyun|kim.taekyung.1|kim.tak-kyeom|kim.taekyun"Kim, D. S."https://zbmath.org/authors/?q=ai:kim.doo-seok|kim.dae-shik|kim.dae-su|kim.dae-sig|kim.david-s|kim.dong-sie|kim.dong-seon|kim.duk-sun|kim.do-sang|kim.dae-seung|kim.dae-san|kim.dong-seong|kim.das-san|kim.dae-seoung|kim.deok-soo|kim.dong-sik|kim.doh-suk|kim.daesoo|kim.dongsoo-s|kim.dong-seoSummary: The aim of this paper is to investigate some properties, recurrence relations and identities involving degenerate Stirling numbers of both kinds associated with degenerate hyperharmonic numbers and also with degenerate Bernoulli, degenerate Euler, degenerate Bell, and degenerate Fubini polynomials.Some identities of fully degenerate dowling and fully degenerate Bell polynomials arising from \(\lambda\)-umbral calculushttps://zbmath.org/1517.110222023-09-22T14:21:46.120933Z"Ma, Yuankui"https://zbmath.org/authors/?q=ai:ma.yuankui"Kim, Taekyun"https://zbmath.org/authors/?q=ai:kim.taekyun"Lee, Hyunseok"https://zbmath.org/authors/?q=ai:lee.hyunseok"Kim, Dae San"https://zbmath.org/authors/?q=ai:kim.dae-san(no abstract)Shape of the asymptotic maximum sum-free sets in integer lattice gridshttps://zbmath.org/1517.110232023-09-22T14:21:46.120933Z"Liu, Hong"https://zbmath.org/authors/?q=ai:liu.hong.1"Wang, Guanghui"https://zbmath.org/authors/?q=ai:wang.guanghui"Wilkes, Laurence"https://zbmath.org/authors/?q=ai:wilkes.laurence"Yang, Donglei"https://zbmath.org/authors/?q=ai:yang.dongleiFor an integer \(n\in \mathbb{N}\), a subset \(A\subset [n]\) is sum-free if it has no solution for the equation \(x+y = z\), i.e., for all \(x, y\in A\) we have \(x+y\notin A\). As a generalization of the definition for sum-free sets, the $(p,q)$-sum free sets and problems associated with the extremal structure are investigated. In this paper, the four authors determine the shape of all sum-free sets in \(\{1,2,\ldots,n\}^2\) of size close to the maximum \(\frac{3}{5}n^2\), solving a problem of \textit{C. Elsholtz} and \textit{L. Rackham} [J. Lond. Math. Soc., II. Ser. 95, No. 2, 353--372 (2017; Zbl 1427.11009)]. They show that all such asymptotic maximum sum-free sets lie completely in the stripe \(\frac{4}{5}n-o(n)\leq x+y\leq \frac{5}{8}n+o(n)\). They also determine for any positive integer \(p\) the maximum size of a subset \(A\subseteq \{1,2,\ldots,n\}^2\) which forbids the triple \((x, y, z)\) satisfying \(px+py=z\).
Reviewer: Weidong Gao (Tianjin)Modular forms and ellipsoidal \(T\)-designshttps://zbmath.org/1517.110332023-09-22T14:21:46.120933Z"Pandey, Badri Vishal"https://zbmath.org/authors/?q=ai:pandey.badri-vishalThe paper under review is a nice application of the modular forms, especially theta functions. \textit{T. Miezaki} [Discrete Math. 313, No. 4, 375--380 (2013; Zbl 1259.05030)] defines spherical \(T\)-design in \(\mathbb{R}^2\). In the paper under review, the author extends this result to special ellipses and the norm form shells for rings of integers of imaginary quadratic fields with class number \(1\). Here the shell means \(\mathbb{Z}^2\)-lattice points with fixed integer norm. The proof is based on calculations with the help of theta functions.
Reviewer: İlker İnam (Bilecik)Optimal strong approximation for quadrics over \(\mathbb{F}_q [t]\)https://zbmath.org/1517.110692023-09-22T14:21:46.120933Z"Sardari, Naser Talebizadeh"https://zbmath.org/authors/?q=ai:talebizadeh-sardari.naser"Zargar, Masoud"https://zbmath.org/authors/?q=ai:zargar.masoudLet \(\mathcal O=\mathbb F_q[t]\) be the polynomial ring over the finite field \(\mathbb F_q\) with \(q\) elements, where \(q\) is a fixed odd prime power, let \(F({\mathbf x})\) be a non-degenerate quadratic form of discriminant \(\Delta\) over \(\mathcal O\) in \(d\geq 4\) variables \(\mathbf{x}=(x_1,\ldots,x_d)\), and let \(f\in \mathcal O\). In this paper, the authors study the optimal strong approximation problem for the quadric \(X_f\) given by the equation \(F(\mathbf{x})=f\). That is, given \(g\in \mathcal O\) and \(\boldsymbol{\lambda} =(\lambda_1,\ldots,\lambda_d)\in \mathcal O^d\), they consider solutions \(\mathbf{x}=(x_1,\ldots,x_d)\in \mathcal O^d\) to the equation \(F(\mathbf{x})=f\) for which \(\mathbf{x}\equiv \boldsymbol{\lambda} \mbox{ mod }g\) (that is, \(x_i \equiv \lambda_i \mbox{ mod }g\) for all \(i=1,\ldots,d\)). Necessary local local conditions for such a solution are that \(X_f(\mathbb F_q((1/t)))\neq \emptyset\) and, for all irreducible \(\varpi \in \mathcal O\), \(F(\mathbf{x})=f\) has a solution \(\mathbf{x}_{\varpi}\in \mathcal O_{\varpi}^d\) such that \(\mathbf{x}_{\varpi} \equiv \boldsymbol{\lambda}\mbox{ mod }\varpi^{\mbox{ord}_{\varpi}(g)}\). The main strong approximation result obtained (Theorem 1.1) is as follows. Let \(\varepsilon >0\) be given, let \(f,g\) be polynomials in \(\mathcal O\) for which the irreducible divisors of \(\Delta\) appear with bounded multiplicity in \(fg\), and let \(\boldsymbol{ \lambda}\in \mathcal O^d\). Suppose that the necessary local conditions for a solution are satisfied. Then there is a constant \(C_{\varepsilon,F}\), independent of \(f\), \(g\) and \(\boldsymbol{\lambda}\), such that if \(d\geq 5\) and \(\mbox{deg }f \geq (4+\varepsilon)\mbox{deg }g+C_{\varepsilon,f}\), then there exists \(\mathbf{x}\in \mathcal O^d\) such that \(F(\mathbf{x})=f\) and \(\mathbf{x}\equiv \boldsymbol{\lambda}\mbox{ mod }g\). In the case \(d=4\), the same result holds with \((4+\varepsilon)\) replaced by \((6+\varepsilon)\).
The method of proof is based on a version of the circle method that was developed by \textit{D. R. Heath-Brown} over the integers [J. Reine Angew. Math. 481, 149--206 (1996; Zbl 0857.11049)] and further developed in a paper of the first author [Duke Math. J. 168, 1887--1927 (2019; Zbl 1443.11030)] to prove an optimal strong approximation result for quadratic forms over the integers. In the present paper, the authors extend the circle method over function fields by proving a stationary phase theorem that allows them to bound certain oscillatory integrals that appear in the circle method. The result obtained is optimal for \(d\geq 5\).
The strong approximation result stated above is used to give a new proof, independent of the Ramanujan conjecture over function fields, that the diameter of a \(k\)-regular Morgenstern Ramanujan graph \(G\) is bounded above by \((2+\varepsilon)\log_{k-1}|G|+O_{\varepsilon}(1)\).
Reviewer: Andrew G. Earnest (Carbondale)On some mock theta functions of order 2 and 3https://zbmath.org/1517.111282023-09-22T14:21:46.120933Z"Kaur, H."https://zbmath.org/authors/?q=ai:kaur.harshil|kaur.harwinder|kaur.harman|kaur.harneet|kaur.harveen|kaur.hargeet|kaur.harmanpreet|kaur.harvinder|kaur.hardish|kaur.harleen|kaur.harpreet|kaur.harsimran|kaur.harvir|kaur.harjeet"Rana, M."https://zbmath.org/authors/?q=ai:rana.mishal|rana.masud|rana.mehwish|rana.meenakshi|rana.mansiAn \((n+t)\)-color partition is a partition of a positive integer \(\nu\), in which a part of size \(n\), \(n\ge0\), can come in \((n+t)\) different colors denoted by \(n_1,n_2,\ldots, n_{n+t}\). Here, zeros are permissible if and only if \(t>0\), and only \(0_t\) is allowed.
In this paper, the authors interpret the following mock theta functions and their generalized version in terms of \((n+t)\)-color partitions:
\begin{align*}
\omega(q) &= \sum_{n=0}^{\infty}\frac{q^{2(n^2+n)}}{(q;q^2)_{n+1}^2},\\
f(q) &= \sum_{n=0}^{\infty}\frac{q^{n^2}}{(-q;q)_{n}^2},\\
A(q) &= \sum_{n=0}^{\infty}\frac{q^{{(n+1)}^2}(-q;q^2)_n}{(q;q^2)_{n+1}^2},\\
B(q) &= \sum_{n=0}^{\infty}\frac{q^{n(n+1)}(-q^2;q^2)_n}{(q;q^2)_{n+1}^2},\\
\mu(q) &= \sum_{n=0}^{\infty}\frac{(-1)^nq^{n^2}(q;q^2)_n}{(-q^2;q^2)_{n}^2}.
\end{align*}
Applying a similar technique they also provide a combinatorial interpretation of two \(q\)-series.
Reviewer: Pankaj Jyoti Mahanta (Lakhimpur)On restricted partitions of numbershttps://zbmath.org/1517.111292023-09-22T14:21:46.120933Z"Mattson, H. F. Jr."https://zbmath.org/authors/?q=ai:mattson.harold-f-junSummary: This paper finds new quasi-polynomials over \(\mathbb{Z}\) for the number \(p_k (n)\) of partitions of \(n\) with parts at most \(k\). Methods throughout are elementary. We derive a small number of polynomials (e.g., one for \(k=3\), two for \(k = 4\) or 5, six for \(k=6)\) that, after addition of appropriate constant terms, take the value \(p_k (n)\). For example, for \(0\leq r<6\) and for all \(q \geq 0, p_3 (6q+r) = p_3 (r)+\pi_0 (q,r)\), a polynomial of total degree 2 in \(q\) and \(r\). In general there are \(M_{\lfloor k/2 \rfloor} = \text{\textsc{lcm}}\{1,2,\ldots,\lfloor k/2 \rfloor\}\) such polynomials. In two variables \(q\) and \(s\), they take the form \(\sum a_{i,j}\left(\begin{smallmatrix} q \\ i \end{smallmatrix}\right) \left(\begin{smallmatrix} s \\ j \end{smallmatrix}\right)\) with \(a_{i,j} \in\mathbb{Z}\), which we call the \textit{proper form} for an integer-valued polynomial. They constitute a quasi-polynomial of period \(M_{\lfloor k/2 \rfloor}\) for the sequence \((p_k (n)-p_k (r))\) with \(n \equiv r \pmod{M_k}\). For each \(k\) the terms of highest total degree are the same in all the polynomials and have coefficients dependent only on \(k\). A second theorem, combining partial fractions and the above approach, finds hybrid polynomials over \(\mathbb{Q}\) for \(p_k (n)\) that are easier to determine than those above. We compare our results to those of Cayley, MacMahon, and Arkin, whose classical results, as recast here, stand up well. We also discuss recent results of Munagi and conclude that circulators in some form are inevitable. At \(k=6\) we find serious errors in Sylvester's calculation of his ``waves''. \textit{J.J. Sylvester} [``On the partition of numbers'', Q. J. Pure Appl. Math. 1, 141-152 (1855)]. The results are generalized to the (not very different) problem called ``making change,'' where significant improvements to existing approaches are found. We find an infinitude of new congruences for \(p_k (n)\) for \(k= 3, 4\), and one new one for \(k=5\). Reduced modulo \(m\) the periodic sequence \((p_k (n))\) is investigated for periodicity and zeros: we find, from scratch, a simple proof of a known result in a special case.A note on Rogers-Ramanujan-Slater type theta function identityhttps://zbmath.org/1517.111302023-09-22T14:21:46.120933Z"Cao, Jian"https://zbmath.org/authors/?q=ai:cao.jian"Arjika, Sama"https://zbmath.org/authors/?q=ai:arjika.sama"Chaudhary, M. P."https://zbmath.org/authors/?q=ai:chaudhary.mahendra-palSummary: In this paper, we research theta function identity involving Rogers-Ramanujan identity and establish a Rogers-Ramanujan-Slater type theta function identity related to \(G(q)\) and \(\varphi(q)\).Weighted words at degree two. I: Bressoud's algorithm as an energy transferhttps://zbmath.org/1517.111312023-09-22T14:21:46.120933Z"Konan, Isaac"https://zbmath.org/authors/?q=ai:konan.isaac\textit{J. Dousse} [Proc. Am. Math. Soc. 145, No. 5, 1997--2009 (2017; Zbl 1357.05010)] introduced a refinement of Siladić's theorem on partitions, where parts occur in two primary and three secondary colors. The proof used the method of weighted words and difference equations. In [Eur. J. Comb. 87, Article ID 103101, 18 p. (2020; Zbl 1439.05025)] the author provided a bijective proof of a generalization of Dousse's theorem from two primary colors to an arbitrary number of primary colors. In this paper, the author gives a generalization of the result given in [loc. cit.], by using the statistic mechanical viewpoint of the integer partitions.
Reviewer: Mircea Merca (Cornu de Jos)Refinement for sequences in partitionshttps://zbmath.org/1517.111322023-09-22T14:21:46.120933Z"Lin, Bernard L. S."https://zbmath.org/authors/?q=ai:lin.bernard-l-s"Lin, Xiaowei"https://zbmath.org/authors/?q=ai:lin.xiaoweiThe \(k\)-measure of a partition \(\lambda\) denoted by \(\mu_k(\lambda)\) is the length of the largest subsequence of parts of \(\lambda\) in which the difference between any two consecutive parts of the subsequence is at least \(k\). It is a new statistic of integer partitions introduced recently by \textit{G. E. Andrews} et al. [Integers 22, Paper A32, 9 p. (2022; Zbl 1494.05007)]. They established a surprising identity that the number of partitions of \(n\) with \(2\)-measure \(m\) is equal to the number of partitions of \(n\) with Durfee square of side \(m\).
In the paper under review, the authors obtain the refinement of the result of Andrews et al., which involves two statistics, the smallest part and the length of partitions. Explicitly, they establish a trivariate generating function identity for partitions counting both the length and the \(2\)-measure, with restriction on the smallest parts and the largest parts. For \(s\geq 0\) and \(N\geq 1\), let \(\mathcal{P}_{s,N}\) be the set of all partitions of \(n\) whose smallest parts are at least \(s+1\) and whose largest parts are at most \(s+N\). Let \(d_N^s(t,z,q)=\sum_{\lambda\in \mathcal{P}_{s,N}}t^{\ell(\lambda)}z^{\mu_2(\lambda)}q^{|\lambda|}\), the identity is:
\[
d_N^s(t,z,q)=\sum_{j\geq 0}\frac{t^jz^jq^{j^2+js}}{(tq^{s+1};q)_j(tq^{s+N-j+1};q)_j} \times\left(\begin{bmatrix}N-j+1 \\ j\end{bmatrix}-tq^{s+N-j+1}\begin{bmatrix}N-j \\ j-1\end{bmatrix}\right).
\]
Letting \(N\rightarrow\infty\) in the above identity, the authors obtain an interesting result which involves a new concept called Durfee rectangle, which is a generalization of the famous Durfee square of partitions. For a nonnegative integer \(d\), the \(d\)-Durfee rectangle of a partition means the largest \(n\times (n+d)\) rectangle that fits in the Ferrers graph, the side of this \(d\)-rectangle is defined as the above largest \(n\).
The interesting result states that the number of partitions of \(n\) with \(\ell\) parts, \(2\)-measure equal to \(j\) and smallest part no less than \(s+1\) is equal to the number of partitions of \(n\) with \(\ell\) parts, \(s\)-Durfee rectangle of side \(j\) and smallest part no less than \(s+1\).
As a corollary of their result, they get the following result which was also established by \textit{G. E. Andrews} et al. [Algebr. Comb. 5, No. 6, 1353--1361 (2022; Zbl 1508.11101)]. That is,
the number of partitions of \(n\) with \(\ell\) parts, \(2\)-measure equal to \(j\) is equal to the number of partitions of \(n\) with \(\ell\) parts, and \(0\)-Durfee square of side \(j\).
Reviewer: Donna Q. J. Dou (Changchun)Neighborly partitions and the numerators of Rogers-Ramanujan identitieshttps://zbmath.org/1517.111332023-09-22T14:21:46.120933Z"Mohsen, Zahraa"https://zbmath.org/authors/?q=ai:mohsen.zahraa"Mourtada, Hussein"https://zbmath.org/authors/?q=ai:mourtada.husseinAmong the most famous and ubiquitous formulas involving \(q\)-series, we find the following two Rogers-Ramanujan identities:
\begin{align*}
\sum_{k=0}^\infty\dfrac{q^{k^2}}{(1-q)(1-q^2)\cdots(1-q^k)} &=\prod_{j=0}^\infty\dfrac{1}{(1-q^{5j+1})(1-q^{5j+4})},\\
\sum_{k=0}^\infty\dfrac{q^{k(k+1)}}{(1-q)(1-q^2)\cdots(1-q^k)} &=\prod_{j=0}^\infty\dfrac{1}{(1-q^{5j+2})(1-q^{5j+3})}.
\end{align*}
In this paper under review, the authors prove two identities which are in some sense dual to the Rogers-Ramanujan identities. These identities are inspired by a correspondence between three kinds of objects, namely, a new type of partitions (neighborly partitions), monomial ideals and some infinite graphs.
Reviewer: Dazhao Tang (Chongqing)The metric chromatic number of zero divisor graph of a ring \(\mathrm{Z_n} \)https://zbmath.org/1517.130062023-09-22T14:21:46.120933Z"Mohammad, Husam Qasem"https://zbmath.org/authors/?q=ai:mohammad.husam-qasem"Ibrahem, Shaymaa Haleem."https://zbmath.org/authors/?q=ai:ibrahem.shaymaa-haleem"Khaleel, Luma Ahmed"https://zbmath.org/authors/?q=ai:khaleel.luma-ahmedSummary: Let \(\Gamma\) be a nontrivial connected graph, \(c:V\left( \Gamma\right)\longrightarrow\mathbb{N}\) be a vertex colouring of \(\Gamma \), and \(L_i\) be the colouring classes that resulted, where \(i=1,2,\dots,k\). A metric colour code for a vertex \(a\) of a graph \(\Gamma\) is \(c\left( a\right)=\left( d \left( a, L_1\right), d \left( a, L_2\right), \dots, d \left( a, L_n\right)\right)\), where \(d\left( a, L_i\right)\) is the minimum distance between vertex \(a\) and vertex \(b\) in \(L_i\). If \(c\left( a\right)\neq c\left( b\right)\), for any adjacent vertices \(a\) and \(b\) of \(\Gamma \), then \(c\) is called a metric colouring of \(\Gamma\) as well as the smallest number \(k\) satisfies this definition which is said to be the metric chromatic number of a graph \(\Gamma\) and symbolized \(\mu\left( \Gamma\right)\). In this work, we investigated a metric colouring of a graph \(\Gamma\left( Z_n\right)\) and found the metric chromatic number of this graph, where \(\Gamma\left( Z_n\right)\) is the zero-divisor graph of ring \(Z_n\).On regularity of Rees algebras of edge ideals of cone graphshttps://zbmath.org/1517.130092023-09-22T14:21:46.120933Z"Nandi, Rimpa"https://zbmath.org/authors/?q=ai:nandi.rimpa"Nanduri, Ramakrishna"https://zbmath.org/authors/?q=ai:nanduri.ramakrishnaSummary: For any simple graph \(G\), let \(K[G]\) denotes the toric algebra of \(G\) over a field \(K\) and \(R(I(G))\) denotes the Rees algebra of the edge ideal \(I(G)\). If \(G\) is a simple graph on \(n\) vertices such that \(G\) has no vertex of degree \(n-1\), then we show that \(\text{reg}(R(I^*))=a_{i_0}(R(I^*))+i_0=a_{i_0}(K[G]\otimes_{R(I(G))} R(I^*))+ i_0\), for some \(i_0\), i.e., the \(=\text{reg}(K[G]\otimes_{R(I(G))}R(I^*))\), regularities are equal and the regularities attain at the same stage, where \(I^*\) denotes the edge ideal of the cone graph of \(G\). We also prove that the bigraded regularities, \(\text{reg}_{(0,1)}(K[G]\otimes_{R(I)}R(I^*))=\text{reg}_{(0,1)}(R(I^*))\) and they attain at the same stage. Finally, we show that \(\text{reg}_{(1,0)}(K[G]\otimes_{R(I)}R(I^*))\le\text{reg}_{(1,0)}(R(I^*))+1\).The saturation number of monomial idealshttps://zbmath.org/1517.130112023-09-22T14:21:46.120933Z"Abdolmaleki, Reza"https://zbmath.org/authors/?q=ai:abdolmaleki.reza"Yazdan Pour, Ali Akbar"https://zbmath.org/authors/?q=ai:yazdan-pour.ali-akbarSummary: Let \(S= \mathbb{K}[x_1,\ldots,x_n]\) be the polynomial ring over a field \(\mathbb{K}\) and \(\mathfrak{m}=(x_1,\ldots,x_n)\) be the homogeneous maximal ideal of \(S\). For an ideal \(I \subset S\), let \(\mathrm{sat}=(I)\) be the minimum number \(k\) for which \(I : \mathfrak{m}^k = I : \mathfrak{m}^{k+1}\). In this paper, we compute the saturation number of irreducible monomial ideals and their powers. We apply this result to find the saturation number of the ordinary powers and symbolic powers of some families of monomial ideals in terms of the saturation number of irreducible components appearing in an irreducible decomposition of these ideals. Moreover, we give an explicit formula for the saturation number of monomial ideals in two variables.Powers of edge ideals of weighted oriented graphs with linear resolutionshttps://zbmath.org/1517.130122023-09-22T14:21:46.120933Z"Banerjee, Arindam"https://zbmath.org/authors/?q=ai:banerjee.arindam.1"Das, Kanoy Kumar"https://zbmath.org/authors/?q=ai:das.kanoy-kumar"Selvaraja, S."https://zbmath.org/authors/?q=ai:selvaraja.sSummary: Let \(\mathcal{D}=(V(\mathcal{D}), E(\mathcal{D}))\) be a weighted oriented graph and \(I(\mathcal{D})\) denote the corresponding edge ideal. In this paper, we give a combinatorial characterization of \(I (\mathcal{D})\) which has a linear resolution. As a consequence, we prove that if \(I(\mathcal{D})\) is the edge ideal of a weighted oriented graph \(\mathcal{D}\), then \(I(\mathcal{D})\) has a linear resolution if and only if all powers of \(I(\mathcal{D})\) have a linear resolution. Also, we prove that if \(\mathcal{D}\) is a weighted oriented graph and \(w(x)>1\) for all \(x \in V(\mathcal{D})\), then \(I(\mathcal{D})\) has a linear resolution if and only if all powers of \(I(\mathcal{D})\) have linear quotients. We provide a lower bound for the regularity of powers of edge ideals of weighted oriented graphs in terms of induced matching. Finally, we obtain a general upper bound for the regularity of edge ideals of weighted oriented graphs.Chordal graphs, higher independence and vertex decomposable complexeshttps://zbmath.org/1517.130142023-09-22T14:21:46.120933Z"Abdelmalek, Fred M."https://zbmath.org/authors/?q=ai:abdelmalek.fred-m"Deshpande, Priyavrat"https://zbmath.org/authors/?q=ai:deshpande.priyavrat"Goyal, Shuchita"https://zbmath.org/authors/?q=ai:goyal.shuchita"Roy, Amit"https://zbmath.org/authors/?q=ai:roy.amit.1"Singh, Anurag"https://zbmath.org/authors/?q=ai:singh.anuragSuppose that \(G\) is a simple graph. Recently, there is a generalization of independent sets in such a way that, for each natural integer \(r\), an \(r\)-independent set in \(G\) is a subset \(A\) of \(V(G)\) such that the induced subgraph of \(G\) on \(A\) has connected components with at most \(r\) vertices. So it is natural to work with \(r\)-independence complex of \(G\), denoted \(\mathrm{Ind}_r(G)\), which is the simplicial complex whose vertices are the vertices of \(G\) and faces are all \(r\)-independent subsets of \(G\), as a generalization of the independence complex.
In the first main result of this paper, it is shown that the \(r\)-independence complex of a tree graph is the independence complex of special chordal hypergraph and so it is shellable by [\textit{R. Woodroofe}, Electron. J. Comb. 18, No. 1, Research Paper P208, 20 p. (2011; Zbl 1236.05213)].
In the second main result, it is shown that the Stanley-Reisner ideal of Alexander dual of \(r\)-independence complex of a caterpillar graph is vertex splittable and so \(r\)-independence complex of such graphs are vertex decomposable by \textit{S. Moradi} and \textit{F. Khosh-Ahang} [Math. Scand. 118, No. 1, 43--56 (2016; Zbl 1339.13010)].
Although it is known that the independence complex of any chordal graph is vertex decomposable and so sequentially Cohen-Macaulay, but it is shown in this paper that there are some chordal graphs such that one of their r-independence complexes is not sequentially Cohen-Macaulay.
Reviewer: Fahimeh Khosh-Ahang Ghasr (Ilam)Line graphs with a Cohen-Macaulay or Gorenstein clique complexhttps://zbmath.org/1517.130152023-09-22T14:21:46.120933Z"Nikseresht, Ashkan"https://zbmath.org/authors/?q=ai:nikseresht.ashkanSummary: Let \(H\) be a simple undirected graph and \(G=\mathrm{L}(H)\) be its line graph. Assume that \(\Delta (G)\) denotes the clique complex of \(G\). We present a complete characterization of those \(H\) for which \(\Delta (G)\) is Cohen-Macaulay or Gorenstein. In addition, we show that if \(\Delta (G)\) is pure, then it is Cohen-Macaulay if and only if it is shellable if and only if it is vertex decomposable.Gorenstein and Cohen-Macaulay matching complexeshttps://zbmath.org/1517.130162023-09-22T14:21:46.120933Z"Nikseresht, Ashkan"https://zbmath.org/authors/?q=ai:nikseresht.ashkanSummary: Let \(H\) be a simple undirected graph. The family of all matchings of \(H\) forms a simplicial complex called the matching complex of \(H\). Here, we give a classification of all graphs with a Gorenstein matching complex. Also we study when the matching complex of \(H\) is Cohen-Macaulay and, in certain classes of graphs, we fully characterize those graphs which have a Cohen-Macaulay matching complex. In particular, we characterize when the matching complex of a graph with girth at least five or a complete graph is Cohen-Macaulay.Algebraic \(h\)-vectors of simplicial complexes through local cohomology. Ihttps://zbmath.org/1517.130172023-09-22T14:21:46.120933Z"Sawaske, Connor"https://zbmath.org/authors/?q=ai:sawaske.connorGiven an infinite field \(k\) and a simplicial complex \(\Delta\), a common theme in studying the \(f\)-vector and \(h\)-vector of \(\Delta\) is the consideration of the Hilbert series of the Stanley-Reisner ring \(k[\Delta]\) modulo a generic linear system of parameters \(\Theta\). Historically, these computations have been restricted to particular classes of complexes (most typically, triangulations of spheres or manifolds). In the paper, for any complex \(\Delta\) of dimension \(d-1\), the author provides a compact topological expression of \(h^a_{d-1}(\Delta)\), where \(h^a_{i}(\Delta)\) is defined to be the dimension of \(k[\Delta]/(\Theta)\) over \(k\) in degree \(i\). In the process, they provide tools and techniques for the possible extension to other coefficients in the Hilbert series.
Reviewer: Tongsuo Wu (Shanghai)Acyclic cluster algebras with dense \(g\)-vector fanshttps://zbmath.org/1517.130202023-09-22T14:21:46.120933Z"Yurikusa, Toshiya"https://zbmath.org/authors/?q=ai:yurikusa.toshiyaSummary: The \(g\)-vector fans play an important role in studying cluster algebras and silting theory. We survey cluster algebras with dense \(g\)-vector fans and show that a connected acyclic cluster algebra has a dense \(g\)-vector fan if and only if it is either finite type or affine type. As an application, we classify finite dimensional hereditary algebras with dense \(g\)-vector fans.
For the entire collection see [Zbl 1516.14004].On generalized Steinberg theory for type AIIIhttps://zbmath.org/1517.140322023-09-22T14:21:46.120933Z"Fresse, Lucas"https://zbmath.org/authors/?q=ai:fresse.lucas"Nishiyama, Kyo"https://zbmath.org/authors/?q=ai:nishiyama.kyoSummary: The multiple flag variety \(\mathfrak{X}=\mathrm{Gr}(\mathbb{C}^{p+q},r)\times (\mathrm{Fl}(\mathbb{C}^p)\times \mathrm{Fl}(\mathbb{C}^q))\) can be considered as a double flag variety associated to the symmetric pair \((G,K)=(\mathrm{GL}_{p+q}(\mathbb{C}),\mathrm{GL}_p(\mathbb{C})\times \mathrm{GL}_q(\mathbb{C}))\) of type AIII. We consider the diagonal action of \(K\) on \(\mathfrak{X}\). There is a finite number of orbits for this action, and our first result is a description of these orbits: parametrization (by a certain set of graphs), dimensions, closure relations and cover relations.
In [Int. Math. Res. Not. 2022, No. 1, 18828--18889 (2022; Zbl 1493.14073)], we defined two generalized Steinberg maps from the \(K\)-orbits of \(\mathfrak{X}\) to the nilpotent \(K\)-orbits in \(\mathfrak{k}\) and those in the Cartan complement of \(\mathfrak{k}\), respectively. The main result in the present paper is a complete, explicit description of these two Steinberg maps by means of a combinatorial algorithm which extends the classical Robinson-Schensted correspondence.Stratification of a singular component of a Springer fiber over \(x^2 = 0\) of the simplest typehttps://zbmath.org/1517.140342023-09-22T14:21:46.120933Z"Mansour, Ronit"https://zbmath.org/authors/?q=ai:mansour.ronit"Melnikov, Anna"https://zbmath.org/authors/?q=ai:melnikov.annaLet \(\mathbb{K}\) be an algebraically closed field and \(x \in M_{n}(\mathbb{K})\) be a nilpotent endomorphism of \(V=\mathbb{K}^{n}\). Let \(\mathcal{F}\) be the variety of full flags over \(V\). The variety \(\mathcal{F}_{x}\) constituted by all the elements \((V_{0}, V_{1}, \ldots, V_{n}) \in \mathcal{F}\) such that \(x(V_{i}) \subset V_{i-1}\) is called a Springer fiber. The components of \(\mathcal{F}_{x}\) are labeled by standard Young tableaux with two columns. If \(T\) is a Young tableau with two columns, one can define a numerical invariant \(\rho(T)\) such that the component \(\mathcal{F}_{T}\) is singular if and only if \(\rho(T) \geq 2\).
In the paper under review authors construct a stratification of \(\mathcal{F}_{T}\) for \(T\) with \(\rho(T)=2\), which reflects also the inner structure of the singular locus, and show that such components have equivalent stratifications if and only if they are isomorphic as algebraic varieties. In addition they provide a combinatorial procedure which partitions the set of such components into the classes of isomorphic components.
Reviewer: Egle Bettio (Venezia)Sublinear circuits and the constrained signomial nonnegativity problemhttps://zbmath.org/1517.140392023-09-22T14:21:46.120933Z"Murray, Riley"https://zbmath.org/authors/?q=ai:murray.riley"Naumann, Helen"https://zbmath.org/authors/?q=ai:naumann.helen"Theobald, Thorsten"https://zbmath.org/authors/?q=ai:theobald.thorstenGiven a finite subset \(\mathcal A\subset \mathbb R^n\), a signomial on \(\mathcal A\) is an expression of the form \(f=\sum_{\alpha\in \mathcal A} c_\alpha e^\alpha\) with \(e^\alpha(x)= \exp(\alpha^T x)\) with real coefficients comprised in \(c=(c_\alpha)_{\alpha\in \mathcal A}\). Polynomials over the positive orthant occur from signomials by a change of variables and choosing \(\mathcal A\subset \mathbb N^n\). As the question of deciding the nonnegativity of \(f\) is known to be NP-hard, one settles for finding sufficient conditions in the coefficients that guarantee nonnegativity. Besides the well known sums-of-squares certificates, certificates based on the arithmetic/geometric mean inequality are increasingly popular. See e.g. [\textit{S. Iliman} and \textit{T. de Wolff}, Res. Math. Sci. 3, Paper No. 9, 35 p. (2016; Zbl 1415.11071)] for early results. The current article investigates this approach in the constrained context. For \(X\subset \mathbb R^n\), convex, \(\beta\in \mathcal A\), define \(C_X(\mathcal A,\beta):=\{\text{signomials nonnegative on \(X\) with at most \(c_\beta\) negative}\}\) and let \(C_X(\mathcal A)=\sum_{\beta\in \mathcal A} C_X(\mathcal A,\beta) \) be the Minkowski sum of the former cones. A cone \(C_X(\mathcal A,\beta)\) is called \(X\)-AGE cone while \(C_X(\mathcal A)\) is an \(X\)-SAGE cone, the acronym obtained from the words `Sum of Arithmetic/Geometric Exponentials'. A deep investigation on cones \(C_X(\mathcal A,\beta)\) and \(C_X(\mathcal A)\) is carried out with some of the important results detailed below.
Let \(\mathbb R^{\mathcal A}\) be the set of all \(|\mathcal A|\)-tuples of reals, indexed by the elements of \(\mathcal A\). For positive \(\nu,c\in \mathcal A\) define the relative entropy \(D(\nu,c)=\sum_{\alpha\in\mathcal A} \nu_\alpha \log \frac{\nu_\alpha}{c_\alpha}\) and let this be \(\infty\) otherwise. Let \(\sigma_X\) be the support function of \(X;\) so \(\sigma_X(y)=\sup\{y^T x: x\in X\}\). Use the letter \(\mathcal A\) also to define a linear operator \(\mathbb R^\mathcal A \stackrel{\mathcal A}{\rightarrow} \mathbb R^n\) by \(\mathcal A \nu= \sum_{\alpha\in \mathcal A } \alpha \nu_\alpha \in \mathbb R^n\). If \(\mu\in \mathbb R^{\mathcal A}\) and \(\beta \in \mathcal A\), let \(\mu_{\setminus\beta}\) the tuple obtained from \(\mu\) by deleting coordinate \(\mu_\beta\). By using that a signomial with at most one negative coefficient is closely related to a convex function, [\textit{R. Murray} et al., Math. Program. Comput. 13, No. 2, 257--295 (2021; Zbl 07447066)] established in their Theorem 1 the following by using the strong duality theorem of convex programming.
Proposition. A signomial \(f= \sum_{\alpha \in \mathcal A} c_\alpha e^\alpha \) belongs to \(C_X(\mathcal A, \beta)\) if and only if there exists a vector \(\nu\in \mathbb R^{\mathcal A}\) which satisfies \(\1^T \nu =0\) and \(\sigma_X(-\mathcal A\nu)+D(\nu_{\setminus \beta}, c_{\setminus \beta} )\leq c_\beta\).
This allows to decide nonnegativity on \(X\) of a signomial with at most one negative coefficient by relative entropy programming. The mentioned proposition concerns \(X\)-AGE cones and it is known that in the unconstrained case every \(f\in C_{\mathbb R^n}(\mathcal A)\) decomposes into AGE-functions supported on singletons and simplicial circuits. The current article undertakes research similar in spirit for the \(X\)-SAGE cones and this requires more than consideration of a signomial's support.
Section 3 introduces the notion of an \(X\)-circuit. This is basic for further analysis. Put \(N_\beta := \{\nu\in \mathbb R^{\mathcal A}: \nu_{\setminus \beta} \geq 0, \1^T\nu=0\}\). A nonzero vector \(\nu^\ast \in N_\beta\) for which \(\sigma_X(-\mathcal A\nu^\ast)<\infty\) is an \(X\)-circuit of \(\mathcal A\) if \(\nu\mapsto \sigma_X(-\mathcal A\nu)\) is strictly sublinear on any line segment containing \(\nu^\ast\) and defined by nonproportional \(\nu^{(1)},\nu^{(2)}\in N_\beta\).
Since in the case that \(X=\mathbb R^n\) one has \(\mathcal A\nu =\mathbb O\), one recovers essentially the notion of a circuit as it occurs in papers as e.g. [ \textit{J. Forsgard} and \textit{T. de Wolff}, SIAM J. Appl. Algebra Geom. 6, No. 3, 468--502 (2022; Zbl 1506.14107); \textit{S. Iliman} and \textit{T. de Wolff}, SIAM J. Optim. 26, No. 2, 1128--1146 (2016; Zbl 1380.12001)]. In Theorem 3.6 vectors \(\nu^\ast\) which are \(X\)-circuits are characterized as edge generators for the cone generated by \(T=\{(\nu, \sigma_X(-\mathcal A\nu)): \nu\in N_\beta, \sigma_X(-\mathcal A\nu)<\infty\}\). From here it is deduced that if convex set \(X\) is polyhedral, it has only finitely many normalized circuits; that is X-circuits \(\lambda\) in which the negative component is \(-1\). Write \(\Lambda_X(\mathcal A,\beta)=\{\lambda\in N_\beta: \lambda \text{ is an \(X\)-circuit for which } \lambda_\beta=-1 \}\) and let \(\Lambda_X(\mathcal A)\) be the set of all normalized \(X\)-circuits of \(\mathcal A\).
In Section 4 it is shown that an X-AGE cone \(C_X(\mathcal A,\beta)\) can be further written as the convex hull of the union of what the authors call \(\lambda\)-witnessed AGE-cones. By definition, given a vector \(\lambda\in N_\beta\) with \(\lambda_\beta=-1\) the \textit{ \(\lambda\)-witnessed AGE-cone} is \(C_X(\mathcal A,\lambda)=\{\sum_{\alpha\in\mathcal A} c_\alpha e^\alpha: \prod_{\alpha\in \lambda^+} \left( \frac{c_\alpha}{\lambda_\alpha} \right)^{\lambda_\alpha} \geq -c_\beta \exp(\sigma_X(-\mathcal A\lambda)), c_{\setminus \beta}\geq \mathbb{O} \}\). The mentioned decomposition, given by Theorem 4.4, says that provided \(\Lambda_X(\mathcal A)\neq \emptyset\), one has \(C_X(\mathcal A,\beta)= \text{conv}\bigcup_{\lambda\in \Lambda_X(\mathcal A,\beta) } C_X(\mathcal A,\lambda)\). The primal power cone associated with a normalized \(X\)-circuit \(\lambda\in \mathbb R^{\mathcal A}\) is given by \(\text{Pow}(\lambda)=\{z\in \mathbb R^{\text{supp}\lambda}: \prod_{\alpha\in \lambda^+} z_\alpha^{\lambda_\alpha} \geq |z_\beta|, z_{\setminus \beta}\geq \mathbb O, \beta=\lambda^- \}\), and its dual has a similar formal aspect and indeed as above formula indicates, \(C_X(\mathcal A,\lambda)\) can be formulated in terms of a dual \(\lambda\)-weighted power cone. From this in Corollary 4.5 it is deduced that if \(X\) is a polyhedron, then \(C_X(\mathcal A)\) is representable as a sum of dual power cones: \(C_X(\mathcal A)=\sum_{\lambda\in \Lambda_X(\mathcal A)} C_X(\mathcal A,\lambda);\) sometimes such cones are even second order representable. A simple representation for the dual cones \(C_X(\mathcal A, \lambda)^*\) is also given.
Section 5 answers the natural question to which extent the \(X\)-circuits generating an \(X\)-SAGE cone are really necessary. In particular if \(X\) is a polyhedron and \(\Lambda_X(\mathcal A)\) is nonempty a minimal set \(\Lambda_X^*(\mathcal A)\) of normalized \(X\)-circuits, called reduced, is identified such that \(C_X(\mathcal A)= \sum_{\lambda\in \Lambda_X^\ast(\mathcal A)} C_X(\mathcal A,\lambda)\). The concept of reducedness is influenced by a percursor of it in [\textit{L. Katthan} et al., Math. Comput. 90, No. 329, 1297--1322 (2021; Zbl 1466.14062)] and by Forsgard and de Wolff's definition [loc.cit] of the Reznick cone. The proof of this needs an extended preparation but some words are dedicated to the question on how to find the reduced \(X\)-circuits (at least in principle). Furthermore as an example the case \(X=[0,\infty[\) is treated in detail making absorption of the theory more palatable. In fact Section 6 adds further details to this example, determining for the case \(\alpha_1<\cdots<\alpha_m\) the extreme rays of \(C_{[0,\infty[}(\{\alpha_1,\dots,\alpha_m\})\).
Section 7 suggests as two lines for future theoretical investigations to formally situate in case of poyhedra \(X\), \(X\)-circuits in the context of matroid theory and an in-depth analysis of multiplicatively convex sets \(S\). These are sets \(S\subset \mathbb R_{>0}^{m} \) for which \(\log S=\{(t_1, \dots, t_m)\in \mathbb R^{m}:(\exp t_1, \dots, \exp t_m)\in S\}\) is convex. Further questions touch more on the usefulness of the investigations for large scale problems. Section 8 is an appendix presenting convex analysis results that were used in some proofs.
The article seems to be very carefully written and the thread of the investigation well motivated.
Reviewer: Alexander Kovačec (Coimbra)The minimum number of multiplicity 1 eigenvalues among real symmetric matrices whose graph is a linear treehttps://zbmath.org/1517.150052023-09-22T14:21:46.120933Z"Ding, Wenxuan"https://zbmath.org/authors/?q=ai:ding.wenxuan"Johnson, Charles R."https://zbmath.org/authors/?q=ai:johnson.charles-richard-jun|johnson.charles-royalThis manuscript deals with the minimum number of eigenvalues with multiplicity 1 among matrices whose associated graph is a linear tree.
A High Degree Vertex (HDV) in a tree is a vertex of degree at least 3, and a tree is called linear if all its HDVs lie on a single induced path of the tree.
Let \(T\) be a linear tree and let \(S(T)\) be the set of real symmetric matrices whose graph is \(T\). If \(U(t)\) is the minimum number of eigenvalues with multiplicity 1 among matrices in \(S(T)\), the authors analyze how \(U(T)\) can change when a vertex is added to \(T\), depending upon how the vertex is added: at an HDV, at a degree 2 vertex, at a pendent vertex, or via edge subdivision. The change is proven to never be by more than 1, but not all such changes can occur. The authors determine the exact set of possibilities.
Specifically, if \(T'\) is a linear tree resulting from the addition of one vertex of \(T\), the authors prove that
\[
|U(T')-U(T)| \leq 1.
\]
They also determine the exact set of possible values of \(U(T')-U(T)\). New bounds and refined bounds are given for \(U(T)\), when \(T\) is a \(k\)-linear tree. In particular, \(U(t) \leq d(T)-k\), where \(d(T)\) denotes the diameter of \(T\), the number of vertices in the longest induced path of \(T\).
Reviewer: Juan Ramón Torregrosa Sánchez (Valencia)Null vectors, Schur complements, and Parter verticeshttps://zbmath.org/1517.150092023-09-22T14:21:46.120933Z"Fallat, Shaun"https://zbmath.org/authors/?q=ai:fallat.shaun-m"Parenteau, Johnna"https://zbmath.org/authors/?q=ai:parenteau.johnnaThis manuscript deals with the inverse eigenvalue problem of matrices whose associated graph is a tree. For a given \(n \times n\) real symmetric matrix \(A=(a_{ij})\), the graph of \(A\), \(G(A)\), is the undirected graph whose vertex set is \(\{1,2,\ldots,n\}\) and the edge set is given by \(\{ij \mid i \neq j \ \mbox{and} \ a_{ij} \neq 0\}\). Given an undirected graph \(G\), \(S(G)\) is the set of real symmetric matrices with the given graph \(G\). Let \(T\) be a tree, let \(v\) be a vertex of \(T\). Any component of the induced subgraph \(T \setminus v\) is called a branch of \(T\) at \(v\). If the degree of \(v\) is \(k\), then there are \(k\) branches of \(T\) at \(v\). If \(T \setminus v=B_1 \cup \cdots \cup B_k\), where \(B_i\) are the branches of \(T\) at \(v\), then for any matrix \(A \in S(T)\), we have \(A(v)=A_1 \oplus \cdots \oplus A_k\), where \(A_i=A(B_i)\).
Eigenvalues of graphs have been a central subject bridging matrix theory and graph theory for many years. Among many other known results, the following ones are fundamental and have been established by \textit{S. Parter} [J. Soc. Ind. Appl. Math. 8, 376--388 (1960; Zbl 0115.24804)] and \textit{G. Wiener} [Linear Algebra Appl. 61, 15--29 (1984; Zbl 0549.15004)]. Suppose that \(T\) is a tree and \(A \in S(T)\). If \(\lambda\) is an eigenvalue of \(A\) with \(m_A(\lambda)\geq 2\) (algebraic multiplicity), then there exists a vertex \(v\) in \(T\) such that:
\begin{itemize}
\item[(1)] \(m_{A(v)}(\lambda)=m_A(\lambda)+1\);
\item[(2)] \(\lambda\) is an eigenvalue of at least three distinct branches of \(T\) at \(v\).
\end{itemize}
In this paper, the authors present an alternative elementary proof of the above results, using a basic linear algebra tool known as the Schur complement of a matrix. This allows them to study the nullspace structure of matrices whose graph is a tree.
Reviewer: Juan Ramón Torregrosa Sánchez (Valencia)Decomposition theorems for involutive solutions to the Yang-Baxter equationhttps://zbmath.org/1517.160302023-09-22T14:21:46.120933Z"Ramírez, S."https://zbmath.org/authors/?q=ai:ramirez.stefanny|ramirez.samuel-a|ramirez.santino|ramirez.sonia-m|ramirez.sara|ramirez.sergio|ramirez.santiago"Vendramin, L."https://zbmath.org/authors/?q=ai:vendramin.lucas|vendramin.leandroAn involutive solution \((X,\sigma,\tau)\) of the Yang-Baxter equation (YBE) is \textit{indecomposable} if the permutation group \(\mathcal{G}(X)=\left\langle \sigma_x\colon x\in X\right\rangle\) of the solution acts transitively on \(X\). And it is \textit{decomposable} otherwise. The \textit{diagonal} of a solution is the permutation \(T\colon X\to X\) defined by \(T(x)=\tau^{-1}_x(x)\).
In 1996 at the International Algebra Conference in Miskolc, Gateva-Ivanova stated the conjecture that all finite involutive square-free solutions of the YBE (of equivalently, solutions with \(T=\mathrm{id}\)) are decomposable. \textit{W. Rump} [Adv. Math. 193, 40--55 (2005; Zbl 1074.81036)] proved the conjecture.
The authors of the paper modify the assumption of Rump's theorem and consider solutions with different structure of the bijection \(T\). In particular, they show that solutions of size \(n>1\) for which \(T\) is a cycle of length \(n\), are indecomposable. But, if the diagonal is a cycle of length \(n-1\), then such solutions are decomposable.
Reviewer: Agata Pilitowska (Warszawa)Orders of units in integral group rings and blocks of defect 1https://zbmath.org/1517.160312023-09-22T14:21:46.120933Z"Caicedo, Mauricio"https://zbmath.org/authors/?q=ai:caicedo.mauricio"Margolis, Leo"https://zbmath.org/authors/?q=ai:margolis.leoLet \(\mathbb Z G\) be the integral group ring of a group \(G\) and \(V(\mathbb Z G)\) be the normalized unit group of \(\mathbb Z G\). The set of the orders of a finite group \(G\) is called the spectrum of \(G\). The so-called Spectrum Problem is the following. Whether the spectra of \(V(\mathbb Z G)\) and \(G\) coincide. The Spectrum Problem has a positive solution for many classes of groups, in particular for the solvable groups [\textit{M. Hertweck}, Commun. Algebra 36, No. 10, 3585--3588 (2008; Zbl 1157.16010)]. The weaker version of the Spectrum Problem is so-called Prime Graph Question: whether \(V(\mathbb Z G)\) and \(G\) have the same prime graph?
The authors prove the following basic result. If \(p\) is a prime and the Sylow \(p\)-subgroup of \(G\) is of order \(p\), then for any prime \(q\), there is an element of order \(pq\) in \(V(\mathbb Z G)\) if and only if there is an element of order \(pq\) in \(G\) (Theorem 1.1).
From this result they answer Prime Graph Question for most sporadic simple groups and some groups of Lie type, including seven new infinite series of such groups, as well as the corresponding almost simple groups. The methods of the indicated results use blocks of cyclic defect, in particular the theory of blocks of defect \(1\) as developed by Brauer, and Young tableaux combinatorics. These serve to restrict the possible actions of a critical unit on \(G\)-modules over fields of positive characteristic. To apply the results to infinitely many groups of Lie type, it is also proven that a product of cyclotomic polynomials of degree at most \(3\) evaluated at primes takes square-free values infinitely many times (this is due to R. Heath-Brown).
The arguments involved can be considered a generalization of the methods involved in earlier papers on the topic, e.g. [\textit{A. Bächle} and \textit{L. Margolis}, Proc. Am. Math. Soc. 147, No. 10, 4221--4231 (2019; Zbl 1444.16052)].
Reviewer: Todor Mollov (Plovdiv)Typed angularly decorated planar rooted trees and generalized Rota-Baxter algebrashttps://zbmath.org/1517.170162023-09-22T14:21:46.120933Z"Foissy, Loïc"https://zbmath.org/authors/?q=ai:foissy.loic"Peng, Xiao-Song"https://zbmath.org/authors/?q=ai:peng.xiao-songA Rota-Baxter algebra is an associative algebra \(A\)\ with a linear endomorphism \(P\), such that for any \(a,b\in A\), we have
\[
P\left(a\right)P\left(b\right)=P\left(aP\left(b\right)\right)+P\left(P\left(a\right)b\right)+\lambda P\left(ab\right)
\]
where \(\lambda\) is a scalar called the weight of the Rota-Baxter operator \(P\). Firstly introduced by \textit{G. Baxter} [Pac. J. Math. 10, 731--742 (1960; Zbl 0095.12705)] in a context of probability theory and popularized by \textit{G.-C. Rota} and \textit{D. A. Smith} [in: Sympos. math. 9, Calcolo Probab., Teor. Turbolenza 1971, 179--201 (1972; Zbl 0255.08003); \textit{G. C. Rota}, Bull. Am. Math. Soc. 75, 325--329 (1969; Zbl 0192.33801)], they now appear in various fields of mathematics and physics. The first appearance of family Rota-Baxter algebra seems to be in [\textit{K. Ebrahimi-Fard} et al., Commun. Math. Phys. 276, No. 2, 519--549 (2007; Zbl 1136.81395)] in the context of renormalization of quantum field theories. This terminology, due to \textit{L. Guo} [J. Algebr. Comb. 29, No. 1, 35--62 (2009; Zbl 1227.05271)], refers to an associative algebra \(A\)\ with a family of linear endomorphisms \(P_{\alpha}:A\rightarrow A\) indexed by the elements of a semigroup \(\left(\Omega,\ast\right)\) such that for any \(a,b\in A\), for any \(\alpha,\beta\in\Omega\),
\[
P_{\alpha}\left(a\right)P_{\beta}\left(b\right)=P_{\alpha\ast\beta}\left(P_{\alpha}\left(a\right)b+aP_{\beta}\left(b\right)+\lambda ab\right)
\]
The notion of matching Rota-Baxter algebra was introduced in [\textit{Y. Zhang} et al., J. Algebra 552, 134--170 (2020; Zbl 1444.16058)], where the Rota-Baxter operators are indexed by the elements of a set \(\Omega\) with no structure, and the weight are given by a family of scales \(\left(\lambda_{\alpha}\right)_{\alpha\in\Omega}\), in which, for any \(a,b\in A\), we have
\[
P_{\alpha}\left(a\right)P_{\beta}\left(b\right) =P_{\beta}\left(P_{\alpha}\left(a\right)\right) b+P_{\alpha}\left(aP_{\beta}\left(b\right)\right) +\lambda_{\beta}P_{\alpha}\left(ab\right)
\]
This paper aims to generalize both family and matching Rota-Baxter algebras in the spirit of what was made in [\textit{L. Foissy}, J. Algebra 586, 1--61 (2021; Zbl 1478.16028)] for dendriform algebras. The authors study the structure needed on the set \(\Omega\) of parameters in order to obtain that free \(\Omega\)-Rota-Baxter algebras are described in terms of typed and angularly decorated planar rooted trees. They also describe free commutative \(\Omega\)-Rota-Baxter algebras generated by a commutative algebra \(A\) in terms of typed words.
Reviewer: Hirokazu Nishimura (Tsukuba)Operator identities on Lie algebras, rewriting systems and Gröbner-Shirshov baseshttps://zbmath.org/1517.170202023-09-22T14:21:46.120933Z"Zhang, Huhu"https://zbmath.org/authors/?q=ai:zhang.huhu"Gao, Xing"https://zbmath.org/authors/?q=ai:gao.xing"Guo, Li"https://zbmath.org/authors/?q=ai:guo.liSummary: Motivated by the pivotal role played by linear operators, many years ago Rota proposed to determine algebraic operator identities satisfied by linear operators on associative algebras, later called Rota's Program on Algebraic Operators. Recent progresses on this program have been achieved in the contexts of operated algebra, rewriting systems and Gröbner-Shirshov bases. These developments also suggest that Rota's insight can be applied to determine operator identities on Lie algebras, and thus to put the various linear operators on Lie algebras in a uniform perspective. This paper carries out this approach, utilizing operated polynomial Lie algebras spanned by nonassociative Lyndon-Shirshov bracketed words. The Lie algebra analog of Rota's program is formulated in terms convergent rewriting systems and equivalently in terms of Gröbner-Shirshov bases. The relation of this Lie algebra analog of Rota's program with Rota's program for associative algebras is established. Applications are given to modified Rota-Baxter operators, differential type operators and Rota-Baxter type operators.Saxl graphs of primitive affine groups with sporadic point stabilizershttps://zbmath.org/1517.200052023-09-22T14:21:46.120933Z"Lee, Melissa"https://zbmath.org/authors/?q=ai:lee.melissa"Popiel, Tomasz"https://zbmath.org/authors/?q=ai:popiel.tomaszA base for a permutation group \(G \le \mathrm{Sym}(\Omega)\) is a subset \(B\subseteq\Omega\) with the property that the pointwise stabilizer of \(B\) in \(G\) is trivial. The \textit{base size} \(b(G)\) of \(G\) is the minimal cardinality of a base for \(G\). Bases have been studied since the late 19-th century [\textit{A. Bochert}, Math. Ann. 33, 584--590 (1889; JFM 21.0141.01)] with particular emphasis on \textit{primitive} groups, namely transitive groups that preserve no non-trivial partition of \(\Omega\), and on groups with small bases. \par If \(G\) has base size \(2\), then the corresponding \textit{Saxl graph} \(\Sigma(G)\) has vertex set \(\Omega\) and two vertices are adjacent if and only if they form a base for \(G\). A recent conjecture of \textit{T. C. Burness} and \textit{M. Giudici} [Math. Proc. Camb. Philos. Soc. 168, No. 2, 219--248 (2020; Zbl 1479.20006)] states that if \(G\) is a finite primitive permutation group with base size \(2\), then \(\Sigma(G)\) has the property that every two vertices have a common neighbour. \textit{T. C. Burness} and \textit{H. Y. Huang} [Algebr. Comb. 5, No. 5, 1053--1087 (2022; Zbl 1511.20012)] have verified that conjecture for almost simple groups with socle \(\mathrm{PSL}(2, q)\) and for almost simple groups with soluble point stabilizers. \par The authors investigate Burness-Giudici conjecture in the case where \(G\) is an affine group and a point stabilizer is an almost quasisimple group whose central quotient is either \(S\) or \(\Aut(S)\) for some sporadic simple group \(S\). Then, the conjecture is verified for all but \(16\) of the groups \(G\).
Reviewer: Marek Golasiński (Olsztyn)Finite simple automorphism groups of edge-transitive mapshttps://zbmath.org/1517.200062023-09-22T14:21:46.120933Z"Jones, Gareth A."https://zbmath.org/authors/?q=ai:jones.gareth-aLet \(\mathcal{G}(T)=\{G\mid G\cong\Aut\mathcal{M}\text{ for some map }\mathcal{M}\in T\}\), where \(T\) is one of the \(14\) Graver-Watkins classes of edge-transitive maps (see [\textit{J. E. Graver} and \textit{M. E. Watkins}, Locally finite, planar, edge-transitive graphs. Providence, RI: American Mathematical Society (AMS) (1997; Zbl 0901.05087)]). Building on earlier results for regular maps and for orientably regular chiral maps [\textit{D. Leemans} and \textit{M. W. Liebeck}, Bull. Lond. Math. Soc. 49, No. 4, 581--592 (2017; Zbl 1378.52013)], it is shown that the non-abelian finite simple groups arising as automorphism groups of maps are those which are not isomorphic to one of the groups listed in the corresponding class:
\begin{itemize}
\item Class \(1\): \(L_{3}(q)\), \(U_{3}(q)\), \(L_{4}(2^{e})\), \(U_{4}(2^{e})\), \(U_{4}(3)\), \(U_{5}(2)\), \(A_{6}\), \(A_{7}\), \(M_{11}\), \(M_{22}\), \(M_{23}\), \(McL\);
\item Classes \(2\), \(2^{\ast}\), \(2P\): \(U_{3}(3)\);
\item Classes \(2\mathrm{ex}\), \(2^{\ast}\mathrm{ex}\), \(2^{P}\mathrm{ex}\): \(L_{2}(q)\), \(L_{3}(q)\), \(U_{3}(q)\), \(A_{7}\);
\item Classes \(3\), \(4\), \(4^{\ast}\), \(4P\): none;
\item Classes \(5\), \(5^{\ast}\), \(5P\): \(L\mathrm{2}(q)\).
\end{itemize}
Reviewer: Wen-Fong Ke (Tainan)On the Weiss conjecture. Ihttps://zbmath.org/1517.200072023-09-22T14:21:46.120933Z"Trofimov, V. I."https://zbmath.org/authors/?q=ai:trofimov.vladimir-iThis is the first article by the author on the Weiss conjecture, going back to work by \textit{R. Weiss} from the late 1970s [Math. Proc. Camb. Philos. Soc. 85, 43--48 (1979; Zbl 0392.20002)]. Suppose that \(\Gamma\) is a connected graph with finite vertex set and that \(G\) is a subgroup of the automorphism group of \(\Gamma\) that acts transitively on the set of vertices. Moreover, suppose that, for all vertices \(x\), the action of the stabiliser \(G_x\) on the set \(\Gamma(x)\) of neighbours of \(x\) in the graph is primitive. Then Weiss' conjecture states that \(|G_x|\) is bounded from above by a number depending only on the number of neighbours of \(x\).
Some parts of the article are fairly general, which makes it a good introduction into the background of Weiss' conjecture. Then the author develops general group theoretic arguments for the structure of the point stabilisers, and later he uses results by \textit{U. Meierfrankenfeld} and \textit{B. Stellmacher} [J. Algebra 351, No. 1, 1--63 (2012; Zbl 1262.20003)], together with the O'Nan-Scott theorem, in order to investigate specific cases of the conjecture. In fact, the author confirms Weiss' conjecture except for the types AS and PA from O'Nan-Scott (as discussed in [\textit{M. W. Liebeck} et al., J. Aust. Math. Soc., Ser. A 44, No. 3, 389--396 (1988; Zbl 0647.20005)]).
Reviewer: Rebecca Waldecker (Halle)Lie elements and the matrix-tree theoremhttps://zbmath.org/1517.200182023-09-22T14:21:46.120933Z"Burman, Yurii"https://zbmath.org/authors/?q=ai:burman.yurii-m"Kulishov, Valeriy"https://zbmath.org/authors/?q=ai:kulishov.valeriySummary: For a finite-dimensional representation \(V\) of a group \(G\) we introduce and study the notion of a Lie element in the group algebra \(k[G]\). The set \(\mathcal{L}(V) \subset k[G]\) of Lie elements is a Lie algebra and a \(G\)-module acting on the original representation \(V\).
Lie elements often exhibit nice combinatorial properties. In particular, we prove a formula, similar to the classical matrix-tree theorem, for the characteristic polynomial of a Lie element in the permutation representation \(V\) of the group \(G = S_n\).Saturated Majorana representations of \(A_{12}\)https://zbmath.org/1517.200192023-09-22T14:21:46.120933Z"Franchi, Clara"https://zbmath.org/authors/?q=ai:franchi.clara"Ivanov, Alexander A."https://zbmath.org/authors/?q=ai:ivanov.alexander-a"Mainardis, Mario"https://zbmath.org/authors/?q=ai:mainardis.marioThe main ingredients of a Majorana representation, as defined by Ivanov, are a finite group \(G\), a \(G\)-stable set \(T\) of involutions in \(G\) and an action of \(G\) on a commutative non-associative algebra \(V\) called Majorana algebra. The motivating example of a Majorana representation is the Monster simple group \(M\), together with its set of Fischer involutions and its action on the Griess algebra.
In the paper under review, the authors investigate the case where \(G\) is the alternating group \(A_{12}\) and \(T\) is the set of involutions of cycle type \(2^2\) or \(2^6\). The authors prove that there is a unique (up to equivalence) corresponding Majorana representation. The relevant Majorana algebra has dimension 3960, and the decomposition of \(V\), viewed as an \(\mathbb{R}A_{12}\)-module, into simple submodules is determined. Thus this Majorana representation comes from an embedding of \(A_{12}\) into the Monster \(M\).
As a consequence of these results, the authors show that the Harada-Norton simple group \(HN\) also has a unique (up to equivalence) Majorana representation. Thus, this similarly comes from an embedding of \(HN\) into \(M\). Some results on Majorana representations of smaller alternating groups are also derived.
Reviewer: Burkhard Külshammer (Jena)The isomorphism of generalized Cayley graphs on finite non-abelian simple groupshttps://zbmath.org/1517.200212023-09-22T14:21:46.120933Z"Zhu, Xiao-Min"https://zbmath.org/authors/?q=ai:zhu.xiaomin"Liu, Weijun"https://zbmath.org/authors/?q=ai:liu.weijun"Yang, Xu"https://zbmath.org/authors/?q=ai:yang.xuSuppose that \(G\) is a finite group, that \(S\) is a subset of \(G\) and that \(\alpha\) is an automorphism of \(G\) of order at most 2 such that the following holds for all \(g,h \in G\):
\(g (g^{-1})^\alpha \notin S\), and if \(g (h^{-1})^\alpha \in S\), then also \(h (g^{-1})^\alpha \in S\).
Then the generalised Cayley graph with respect to \(G\), \(S\) and \(\alpha\), denoted by \(GC(G, S, \alpha)\), has vertex set \(G\) and edge set \(\{\{g, sg^\alpha\}\mid g \in G, s \in S\}\).
In the special case where \(\alpha\) is the identity map and \(S\) is a generating set, this is the well-known Cayley graph, but the definition is more general and allows for \(G\) not to be generated by \(S\).
The paper focuses on a generalisation of so-called \(m\)-CI-groups, where \(m \in \mathbb{N}\). \(G\) is an \(m\)-CI-group if and only if, for all inverse-closed subsets \(S\) of \(G\) of size at most \(m\), the resulting Cayley graphs are isomorphic.
In the present article, this notion is extended for generalised Cayley graphs. The authors investigate, for finite simple non-abelian groups \(G\), how if \(S_1\) and \(S_2\) are small subsets and if the graphs \(GC(G, S_1, \alpha_1)\) and \(GC(G, S_2, \alpha_2)\) are isomorphic, this gives information about the subsets and the automorphisms. The article also gives some examples, discusses related questions and gives an overview over previous results in this area.
Reviewer: Rebecca Waldecker (Halle)\text{M}, \text{B} and \(\mathrm{Co}_1\) are recognisable by their prime graphshttps://zbmath.org/1517.200232023-09-22T14:21:46.120933Z"Lee, Melissa"https://zbmath.org/authors/?q=ai:lee.melissa"Popiel, Tomasz"https://zbmath.org/authors/?q=ai:popiel.tomaszThe prime graph or Kegel-Gruenberg graph \(\Gamma(G)\) of a finite group \(G\) is the graph whose vertices are the prime divisors of \(\lvert G\rvert\) and whose edges are the pairs \(\{p,q\}\) for which \(G\) possesses an element of order \(pq\). We say that a group is \textit{recognisable} by its prime graph if every group \(H\) such that \(\Gamma(H)=\Gamma(G)\) is isomorphic to \(G\).
The problem of recognisability by the prime graph of the sporadic finite simple groups has been addressed by several authors, and there were only three cases left: the Monster \(M\), the Baby Monster \(B\), and the first Conway group \(\operatorname{Co}_1\). In this paper, the authors prove that these three groups are recognisable by their prime graphs.
Reviewer: Ramón Esteban-Romero (València)On the soluble graph of a finite grouphttps://zbmath.org/1517.200242023-09-22T14:21:46.120933Z"Burness, Timothy C."https://zbmath.org/authors/?q=ai:burness.timothy-c"Lucchini, Andrea"https://zbmath.org/authors/?q=ai:lucchini.andrea"Nemmi, Daniele"https://zbmath.org/authors/?q=ai:nemmi.danieleGiven an insoluble group \(G\) with soluble radical \(R(G)\), we can consider a graph whose vertices are the elements of \(G\setminus R(G)\) and in which two vertices \(x\) and \(y\) are adjacent if and only if they generate a soluble subgroup of \(G\). This graph is called the \textit{soluble graph} of \(G\). This construction is a generalisation of the commutativity graph of \(G\), in which two elements are adjacent if, and only if, they commute, that is, they generate an abelian subgroup. The main result of this paper states that this graph is always connected and its diameter \(\delta_{\mathcal{S}}(G)\) is at most~\(5\). The authors prove some more accurate results in Theorem 2:
\begin{itemize}
\item[1.] If \(G\) is not almost simple, then \(\delta_{\mathcal{S}}(G)\le 3\).
\item[2.] If \(G\) is almost simple with socle \(G_0\), then \(\delta_{\mathcal{S}}(G)\le 5\). In addition:
\begin{itemize}
\item[(a)] If \(G_0=\operatorname{L}_2(q)\) and \(q\ge 8\), then \(\delta_{\mathcal{S}}(G)=2\) if \(\operatorname{PGL}_2(q)\le G\), otherwise, \(\delta_{\mathcal{S}}(G)=3\).
\item[(b)] If \(G_0=A_n\) and \(n\ge 7\), then either \(\delta_{\mathcal{S}}(G)=3\) or \(G=A_n\) and \(n\in \{p, p+1\}\), where \(p\) is a prime with \(p\equiv 3\pmod{4}\).
\item[(c)] If
\begin{align*}
\mathcal{A}&=\{A_{11}, A_{12}, \operatorname{L}_5^\epsilon(2), \operatorname{M}_{12}, \operatorname{M}_{22}, \operatorname{M}_{23}, \operatorname{M}_{24}, \operatorname{HS}, \operatorname{J}_3\},\\
\mathcal{B}&=\{A_n, \operatorname{L}_7^\epsilon(2), E_6(2), \operatorname{Co}_2, \operatorname{Co}_3, \operatorname{McL}, \mathbb{B}\},
\end{align*}
and \(G\in\mathcal{A}\cup \mathcal{B}\), then \(\delta_{\mathcal{S}}(G)\ge 4\), with equality if \(G\in\mathcal{A}\).
\item[(d)] If \(G=G_0\) is not isomorphic to a classical group, then \(\delta_{\mathcal{S}}(G)\ge 3\).
\end{itemize}
\end{itemize}
Recall that a natural number \(p\) is a \emph{Sophie Germain prime} if both \(p\) and \(2p+1\) are prime numbers. Theorem~4 states that if \(p\ge 5\) is a Sophie Germain prime, then \(\delta_{\mathcal{S}}(A_{2p+1})\in\{4,5\}\). For finite insoluble groups \(G\) with \(R(G)=1\) and \(x\in G\) nontrivial, Theorem~5 says that we have that either there exists an involution \(y\) with \(\delta(x,y)\le 2\), or \(G\) is the Mathieu group \(\operatorname{M}_{23}\) and \(x\) is of order~\(23\).
Some other properties of this graph and other related graphs, like the ones obtained by replacing soluble by nilpotent, metabelian or metacyclic, are shown in this paper.
The techniques used to prove the results include a reduction to almost simple groups and the classification of almost simple groups. Some \textsc{Magma} algorithms have also been used.
Section~8 of the paper includes some open problems and conjectures. The existence of groups \(G\) with \(\delta_{\mathcal{S}}(G)=5\) is left as an open problem, as well as the existence of infinitely many groups with \(\delta_{\mathcal{S}}(G)\ge 4\). The existence of infinitely many Sophie Germain primes would give a positive answer to this last problem, as well as if for all primes \(p\ge 11\) with \(p\equiv 3\pmod{4}\) we had \(\delta_{\mathcal{S}}(A_p)\ge 4\). The determination of all simple groups with \(\delta_{\mathcal{S}}(G)=2\) is also left as an open problem, as well as to know whether \(\delta_{\mathcal{S}}(G)\le 3\) for every non-simple group~\(G\). Another open problem are to understand the relation between \(\delta_{\mathcal{S}}(G)\) and \(\delta_{\mathcal{S}}(G_0)\) if \(G\) is a finite almost simple group with socle \(G_0\) and to understand the relation between \(\delta_{\mathcal{S}}(G)\) and \(\delta_{\mathcal{S}}(T)\) when \(G\) is a monolithic group with socle \(T^k\), where \(T\) is a non-abelian finite simple group and \(k\ge 2\).
Reviewer: Ramón Esteban-Romero (València)Skew product groups for monolithic groupshttps://zbmath.org/1517.200292023-09-22T14:21:46.120933Z"Bachratý, Martin"https://zbmath.org/authors/?q=ai:bachraty.martin"Conder, Marston"https://zbmath.org/authors/?q=ai:conder.marston-d-e"Verret, Gabriel"https://zbmath.org/authors/?q=ai:verret.gabrielThe main result of the paper is a classification of the finite groups \(G\) having a complementary factorization \(G=B\cdot C\) (i.e. \(B\), \(C\) are subgroups of \(G\) with \(B\cap C=\{1\}\)), where \(B\) is monolithic (i.e. \(B\) has a unique minimal normal subgroup and it is non-abelian), \(C\) is non-trivial cyclic and core-free in \(G\) (i.e. \(C\) contains no non-trivial normal subgroup of \(G\)). As a consequence, it turns out that finite nonabelian simple groups rarely have skew morphisms that are not automorphisms. Recall that a \textit{skew morphism} of a group \(B\) is a bijection \(\varphi:B\to B\), \(\varphi(1)=1\), for which there exists a map \(\pi:B\to \mathbb{N}\) such that for all \(a,b\in B\) we have \(\varphi(a\cdot b)=\varphi(a)\cdot \varphi^{\pi(a)}(b)\). Note that skew morphisms of \(B\) are closely related to complementary factorizations of the form \(G=B\cdot C\) where \(C\) is cyclic and core-free in \(G\). Skew morphisms were introduced in the study of regular Cayley maps (certain embeddings of Cayley graphs into surfaces).
Reviewer: Matyas Domokos (Budapest)Extending results of Morgan and Parker about commuting graphshttps://zbmath.org/1517.200342023-09-22T14:21:46.120933Z"Beike, Nicolas F."https://zbmath.org/authors/?q=ai:beike.nicolas-f"Carleton, Rachel"https://zbmath.org/authors/?q=ai:carleton.rachel"Costanzo, David G."https://zbmath.org/authors/?q=ai:costanzo.david-g"Heath, Colin"https://zbmath.org/authors/?q=ai:heath.colin"Lewis, Mark L."https://zbmath.org/authors/?q=ai:lewis.mark-l"Lu, Kaiwen"https://zbmath.org/authors/?q=ai:lu.kaiwen"Pearce, Jamie D."https://zbmath.org/authors/?q=ai:pearce.jamie-dLet \(G\) be a finite group; the commuting graph of \(G\) is the graph \(\Gamma (G)\) whose vertex set is \(G\setminus Z(G)\), that is the set of all non central elements of \(G\), and two vertices \(x\) and \(y\) are adjacent if and only if \([x,y]=1\), that is if and only if they commute.
Commuting graphs were studied by many authors; much of the research regarding this subject is related to simple groups. For instance, \textit{R. Solomon} and \textit{A. Woldar} [J. Group Theory 16, 793--824 (2013; Zbl 1293.20024)] proved that simple groups are characterized by their commuting graph.
\textit{A. Iranmanesh} and \textit{A. Jafarzadeh} [Acta Math. Acad. Paedagog. Nyházi. (N.S.) 23, No. 1, 7--13 (2007; Zbl 1135.20014)] conjectured that there is a universal bound on the diameter of commuting graphs but \textit{M. Giudici} and \textit{C. Parker} [J. Comb. Theory, Ser. A 120, No. 7, 1600--1603 (2013; Zbl 1314.05055)] constructed a family of 2-groups, nilpotent of class 2, for which there is no bound on the diameter of the commuting graphs.
On the other hand, \textit{C. Parker} [Bull. Lond. Math. Soc. 45, No. 4, 839--848 (2013; Zbl 1278.20017)] proved that the commuting graph of a solvable group \(G\) with trivial center is disconnected if and only if \(G\) is a Frobenius group or it has normal subgroups \(K\le L\) such that \(L\) and \(G/K\) are Frobenius groups with Frobenius kernels \(K\) and \(L/K\), respectively (that is it is a 2-Frobenius group). Moreover, when \(\Gamma (G)\) is connected it has diameter at most 8.
Afterwards \textit{G. L. Morgan} and \textit{C. W. Parker} [J. Algebra 393, 41--59 (2013; Zbl 1294.20033)] removed the solvability hypothesis on \(G\) and proved that if \(G\) is any group with trivial center, then all the connected components of \(\Gamma(G)\) have diameter at most 10.
In this paper, the authors try to extend results of Parker [loc. cit,] and Parker and Morgan [loc. cit,] and they show that it is possible to replace the hypothesis that \(Z(G)=1\) with the hypothesis that \(G'\cap Z(G)=1\). The main result is the following
Theorem 1.1 Let \(G\) be a group and suppose that \(G'\cap Z(G)=1\); then
1) \(\Gamma(G)\) is connected if and only if \(\Gamma (G/Z(G))\) is connected;
2) every connected component of \(\Gamma(G)\) has diameter at most \(10\);
3) if \(G\) is solvable and \(\Gamma (G)\) is connected, then \(\Gamma (G)\) has diameter at most \(8\);
4) if \(G\) is solvable, then \(\Gamma (G)\) is disconnected if and only if \(G/Z(G)\) is either a Frobenius group or a \(2\)-Frobenius group.
The hypothesis that \(G'\cap Z(G)=1\) can be relaxed, in fact, it suffices to assume that, for all \(x,y\in G\), the commutator \([x,y]\in Z(G)\) if and only if \([x,y]=1\).
A class of finite groups satisfying the hypothesis of Theorem 1.1 is, for instance, the class of \(A\)-groups. In fact, if every Sylow subgroup of a group \(G\) is abelian (that is \(G\) is an \(A\)-group), then it is possible to prove that \(G'\cap Z(G)=1\)
Finally, the authors present some examples to clarify various points.
Reviewer: Chiara Nicotera (Salerno)Finite groups whose commuting conjugacy class graphs have isolated verticeshttps://zbmath.org/1517.200362023-09-22T14:21:46.120933Z"Saeidi, Amin"https://zbmath.org/authors/?q=ai:saeidi.aminIn this paper, the author studies the so-called \textit{commuting conjugacy class graph} of a finite group \(G\), denoted \(\Gamma(G)\), introduced by \textit{M. Herzog} et al. in [Commun. Algebra 37, No. 10, 3369--3387 (2009; Zbl 1187.20027)]. The vertex set of this graph is the set of non-identity conjugacy classes of \(G\), and two conjugacy classes \( C_1\) and \(C_2\) are adjacent if and only if we can find \(x \in C_1\) and \(y \in C_2\) such that \(xy =yx\). The vertex of the graph corresponding to the conjugacy class of an element \(x \in G\) is denoted by \([x]\).
Groups whose corresponding graphs contain isolated vertices are investigated in the paper under review. It is shown that if \(\Gamma(G)\) has an isolated vertex \([x]\), then the order of \(x\) must be a prime number \(p\), and in such case it is said that the group \(G\) \textit{satisfies the \(p\)-isolation property}. Moreover, the Sylow \(p\)-subgroups of such group \(G\) are \(CC\)-subgroups, i.e., for every non-trivial \(x \in P\), \(P\) a Sylow \(p\)-subgroup of \(G\), it holds \(C_G(x) \leq P\).
A complete classification of solvable groups with the \(p\)-isolation property, \(p\) a prime, is given in the paper (Theorem 1). It is also proved that if \(G\) is nonsolvable, then either \(G/F(G)\) is almost simple, or \(G\) is sharply \(2\)-transitive (Theorem 2), where \(F(G)\) denotes the Fitting subgroup of \(G\).
Since the structure of finite non-solvable sharply \(2\)-transitive groups is well-known (they are Frobenius groups with a restricted Frobenius complement), the next problem is to determine almost simple groups satisfying the \(p\)-isolation property. This problem has been solved in the paper for almost simple groups with sporadic socles (Theorem 3), as well as for certain families of finite simple groups (Theorems 3.3 and 3.7).
Moreover, a complete classification of non-solvable groups satisfying the \(p\)-isolation property is given for the primes \(p = 2\) and \(p = 3\) (Propositions 3.6 and 3.10).
Reviewer: Ana Martínez-Pastor (València)Cogrowth series for free products of finite groupshttps://zbmath.org/1517.200382023-09-22T14:21:46.120933Z"Bell, Jason"https://zbmath.org/authors/?q=ai:bell.jason-p"Liu, Haggai"https://zbmath.org/authors/?q=ai:liu.haggai"Mishna, Marni"https://zbmath.org/authors/?q=ai:mishna.marniLet \(G\) be a group with finite generating set \(S\). Let \(\mathcal{L}(G,S)\) be the set of elements in the free monoid \(S^{*}\) of \(S\) whose image in \(G\) is the identity. For each \(n\geq 0\), let \(CL(n,G,S)\) be the number of words of length \(n\) which are equal to the identity of \(G\). This is the cogrowth function of \(G\) and generating set \(S\). The generating function for this sequences is the cogrowth series and is given by \(F_{G,S}(t)=\sum_{n\geq 0}CL(n,G,S)t^{n}\). This paper studies these series for virtually free groups. The authors provide explicit formulae and several examples. The main results are as follows: Consider \(G_{1}, \ldots ,G_{r}\) finite groups with generating sets \(S_{1},\ldots, S_{r}\), respectively. For each \(i=1,\dots,r\), let \(G_{i}^{*m_{i}}\) denote the \(m_{i}\)-times free product of \(G_{i}\) and let \(S_{i}^{(j)}\), \(j=1,\ldots ,m_{i}\) be copies of \(S_{i}\) in the corresponding copies of \(G_{i}\). Moreover, let \(G=G_{1}^{*m_{1}}*\cdots *G_{r}^{*m_{r}}\) and \(S=\bigcup_{i=1}^{r}\bigcup_{j=1}^{m_{i}}S_{i}^{(j)}\). The authors prove:
Theorem. Let \(G\) and \(S\) be as above. Then the cogrowth series \(F(t)=F_{G,S}(t)\) is algebraic and satisfies \(\Lambda(t,F(t))=0\), where \(\Lambda(t,z)\) is a non-zero polynomial with rational coefficients satisfying
\[
\deg_{t}(\Lambda), \deg_{z}(\Lambda)\leq (\prod_{i=1}^{r}\Delta_{i})(1+\sum_{i=1}^{r}\frac{1}{\Delta_{i}}),
\]
where \(\Delta_{i}\) is the sum of the degrees of the irreducible representations of \(G_{i}\), \(i=1,\ldots , r\).
In the second result, the authors provide explicit expressions for the cogrowth series for the following groups with specific generating sets: \(\mathbb{Z}/d*\cdots *\mathbb{Z}/d\), the \(m\)-times free product of the finite cyclic group \(\mathbb{Z}/d\) for \(d,m\geq 2\), the \(m\)-times free product of \(\mathbb{Z}/2\) free product with a free group of rank \(s\), \(m,s\geq 0\) and \(\mathbb{Z}/2*\mathbb{Z}/n\), \(n\geq 2\).
Lastly, the authors find a gap in the radius of convergence of the generating series of the cogrowth series for a group \(G\) with generating set \(S\).
Reviewer: Daniel Juan Pineda (Michoacán)Invariants for metabelian groups of prime power exponent, colorings, and stairshttps://zbmath.org/1517.200602023-09-22T14:21:46.120933Z"Barmak, Jonathan Ariel"https://zbmath.org/authors/?q=ai:barmak.jonathan-arielIn this paper, the authors study the free metabelian group \(M(2,n)\) of prime power exponent \(n\) in two generators, that is, the quotient of the Burnside group \(B(2,n)\) by its second derived subgroup. Contrary to Burnside groups, the groups \(M(2, n)\) are known to be finite for every \(n\). However, their order has been determined in very few cases if \(n\) is not prime. The authors define invariants (group homomorphisms) \(M(2,n)' \rightarrow \mathbb{Z}_{n}\) (that they construct from colorings of the squares in the integer grid \(\mathbb{R} \times \mathbb{Z} \cup \mathbb{Z} \times \mathbb{R}\)) and use them to prove that certain identities do not hold in \(M(2, n)\) and to improve bounds for its order obtained by \textit{M. F. Newman} [Lect. Notes Math. 1098, 87--98 (1984; Zbl 0566.20016)]. They study identities in \(M(2,n)\), which give information about identities in the Burnside group \(B(2,n)\) and the restricted Burnside group \(R(2,n)\).
Reviewer: M. Carmen Pedraza-Aguilera (València)On automorphisms of undirected Bruhat graphshttps://zbmath.org/1517.200632023-09-22T14:21:46.120933Z"Gaetz, Christian"https://zbmath.org/authors/?q=ai:gaetz.christian"Gao, Yibo"https://zbmath.org/authors/?q=ai:gao.yiboIn this paper, the authors study automorphisms of the undirected Bruhat graph and classify when the undirected Bruhat graph is vertex-transitive.
Reviewer: Chen Sheng (Harbin)On the monoid of partial isometries of a finite star graphhttps://zbmath.org/1517.200892023-09-22T14:21:46.120933Z"Fernandes, Vítor H."https://zbmath.org/authors/?q=ai:fernandes.vitor-h|fernandes.vitor-hugo"Paulista, Tânia"https://zbmath.org/authors/?q=ai:paulista.taniaLet \(S_n =(\{0, 1,\ldots, n-1\},\{\{0,i\}\mid i=1, \ldots, n-1\})\) be the \textit{star graph} with \(n\) vertices and \(\mathcal{DP}S_n\) the monoid of all partial isometries of \(S_n\) (distance preserving partial transformations of \(S_n\)). In this paper, the cardinal of \(\mathcal{DP}S_n\) is calculated, Green's relations and generating sets of \(\mathcal{DP}S_n\) are described and it is proved that the rank (the minimum size of a generating set) of \(\mathcal{DP}S_n\) is 3, for \(n=3\), and 5, for \(n\geq 4\). The main theorem states that for \(n\geq 4\) the presentation of a monoid \(\mathcal{DP}S_n\) is defined by \(3n+9\) relations, all explicitly described.
Reviewer: Peeter Normak (Tallinn)Connections between vector-valued and highest weight Jack and Macdonald polynomialshttps://zbmath.org/1517.330072023-09-22T14:21:46.120933Z"Colmenarejo, Laura"https://zbmath.org/authors/?q=ai:colmenarejo.laura"Dunkl, Charles F."https://zbmath.org/authors/?q=ai:dunkl.charles-f"Luque, Jean-Gabriel"https://zbmath.org/authors/?q=ai:luque.jean-gabrielSummary: We analyze conditions under which a projection from the vector-valued Jack or Macdonald polynomials to scalar polynomials has useful properties, specially commuting with the actions of the symmetric group or Hecke algebra, respectively, and with the Cherednik operators for which these polynomials are eigenfunctions. In the framework of representation theory of the symmetric group and the Hecke algebra, we study the relation between singular nonsymmetric Jack and Macdonald polynomials and highest weight symmetric Jack and Macdonald polynomials. Moreover, we study the quasistaircase partition as a continuation of our study on the conjectures of Bernevig and Haldane on clustering properties of symmetric Jack polynomials.The heat kernel on the diagonal for a compact metric graphhttps://zbmath.org/1517.340382023-09-22T14:21:46.120933Z"Borthwick, David"https://zbmath.org/authors/?q=ai:borthwick.david"Harrell, Evans M. II"https://zbmath.org/authors/?q=ai:harrell.evans-m-ii"Jones, Kenny"https://zbmath.org/authors/?q=ai:jones.kennySummary: We analyze the heat kernel associated with the Laplacian on a compact metric graph, with standard Kirchhoff-Neumann vertex conditions. An explicit formula for the heat kernel as a sum over loops, developed by Roth and Kostrykin-Potthoff-Schrader, allows for a straightforward analysis of small-time asymptotics. We show that the restriction of the heat kernel to the diagonal satisfies a modified version of the heat equation. This observation leads to an ``edge'' heat trace formula, expressing the a sum over eigenfunction amplitudes on a single edge as a sum over closed loops containing that edge. The proof of this formula relies on a modified heat equation satisfied by the diagonal restriction of the heat kernel. Further study of this equation leads to explicit formulas for graphs which are symmetric about each vertex.An uncountable ergodic Roth theorem and applicationshttps://zbmath.org/1517.370062023-09-22T14:21:46.120933Z"Durcik, Polona"https://zbmath.org/authors/?q=ai:durcik.polona"Greenfeld, Rachel"https://zbmath.org/authors/?q=ai:greenfeld.rachel"Iseli, Annina"https://zbmath.org/authors/?q=ai:iseli.annina"Jamneshan, Asgar"https://zbmath.org/authors/?q=ai:jamneshan.asgar"Madrid, José"https://zbmath.org/authors/?q=ai:madrid.jose-a-jimenezAmenable group actions, Følner sequences, and recurrence theorems are fundamental concepts in the study of group actions and dynamics. An amenable group is a group that possess certain properties, such as the existence of a left-invariant mean on its left regular representation. Følner sequences, on the other hand, are sequences of finite subsets of a group that exhibit a balanced growth property. These sequences play a crucial role in establishing key results in the theory of amenable group actions, particularly in proving various forms of recurrence theorems.
The paper studies extensions, uniformity and combinatorial implications related to Furstenberg's double recurrence theorem. This theorem, which is closely connected to Roth's theorem, plays a central role throughout the paper.
The authors prove an uncountable version of the ergodic Roth theorem of \textit{V. Bergelson} et al. [Am. J. Math. 119, No. 6, 1173--1211 (1997; Zbl 0886.43002)]
for discrete amenable groups. More specifically, they show that if \(\Gamma\) is an arbitrary amenable discrete group then for every Følner net a certain ergodic average in an abstract Roth \(\Gamma\)-dynamical system converges, and, this situation is independent of the choice of the Følner net. The group is not assumed to be countable, and the space is not necessarily separable. Then the authors use this theorem to obtain a syndetic subset of \(\Gamma.\) Recall that a subset of a group or semigroup is said to be syndetic if it has bounded gaps between its elements. Syndetic sets are closely related to multiple recurrence and multiple ergodic averages. Their presence or absence has important implications for the behavior of ergodic averages and the convergence of related theorems. The syndetic subset obtained by the authors are used for the syndeticity of multiple return times for an ultralimit system. By using an uncountable version of the Furstenberg correspondence principle they prove a combinatorial theorem about syndetic sets and triangular patterns in certain subsets of the Cartesian square of arbitrary amenable groups.
One of the main results of the paper is the syndeticity of double return times for arbitrary amenable groups by extending the double recurrence theorem of \textit{V. Bergelson} et al. [loc. cit.] to uncountable amenable groups acting on arbitrary not necessarily separable spaces. Further they obtain a new uniformity aspect for the set of double return times in the amenable Roth theorem. They generalize this to Roth-type \(\Gamma\)-measure-preserving dynamical systems where \(\Gamma\) is uniformly amenable. They show the existence of a lower bound on the degree of syndeticity uniformly over a class of Roth-type measure-preserving dynamical systems for a uniformly amenable set of groups. To do this they use lower Banach densities defined on subsets of the discrete group \(\Gamma\) and they consider ultraproducts of measure-preserving dynamical systems.
Reviewer: Nazife Erkursun Ozcan (Ankara)Variants of \(q\)-hypergeometric equationhttps://zbmath.org/1517.390022023-09-22T14:21:46.120933Z"Hatano, Naoya"https://zbmath.org/authors/?q=ai:hatano.naoya"Matsunawa, Ryuya"https://zbmath.org/authors/?q=ai:matsunawa.ryuya"Sato, Tomoki"https://zbmath.org/authors/?q=ai:sato.tomoki"Takemura, Kouichi"https://zbmath.org/authors/?q=ai:takemura.kouichiSummary: We introduce two variants of the \(q\)-hypergeometric equation. We obtain several explicit solutions of the variants of the \(q\)-hypergeometric equation. We show that a variant of the \(q\)-hypergeometric equation can be obtained by a restriction of the \(q\)-Appell equation of two variables.Multiplicative and semi-multiplicative functions on non-crossing partitions, and relations to cumulantshttps://zbmath.org/1517.460502023-09-22T14:21:46.120933Z"Celestino, Adrian"https://zbmath.org/authors/?q=ai:celestino.adrian"Ebrahimi-Fard, Kurusch"https://zbmath.org/authors/?q=ai:ebrahimi-fard.kurusch"Nica, Alexandru"https://zbmath.org/authors/?q=ai:nica.alexandru"Perales, Daniel"https://zbmath.org/authors/?q=ai:perales.daniel"Witzman, Leon"https://zbmath.org/authors/?q=ai:witzman.leonSummary: We consider the group \((\mathcal{G}, \ast)\) of unitized multiplicative functions in the incidence algebra of non-crossing partitions, where ``\( \ast \)'' denotes the convolution operation. We introduce a larger group \(( \widetilde{\mathcal{G}}, \ast)\) of unitized functions from the same incidence algebra, which satisfy a weaker \textit{semi-multiplicativity} condition. The natural action of \(\widetilde{\mathcal{G}}\) on sequences of multilinear functionals of a non-commutative probability space captures the combinatorics of transitions between moments and some brands of cumulants that are studied in the non-commutative probability literature. We use the framework of \(\widetilde{\mathcal{G}}\) in order to explain why the multiplication of free random variables can be very nicely described in terms of Boolean cumulants and more generally in terms of \(t\)-Boolean cumulants, a one-parameter interpolation between free and Boolean cumulants arising from work of \textit{M. Bożejko} and \textit{J. Wysoczański} [Ann. Inst. Henri Poincaré, Probab. Stat. 37, No. 6, 737--761 (2001; Zbl 0995.60004)]. It is known that the group \(\mathcal{G}\) can be naturally identified as the group of characters of the Hopf algebra Sym of symmetric functions. We show that \(\widetilde{\mathcal{G}}\) can also be identified as group of characters of a Hopf algebra \(\mathcal{T} \), which is an incidence Hopf algebra in the sense of Schmitt. Moreover, the inclusion \(\mathcal{G} \subseteq \widetilde{\mathcal{G}}\) turns out to be the dual of a natural bialgebra homomorphism from \(\mathcal{T}\) onto Sym.Infinite quantum permutationshttps://zbmath.org/1517.460522023-09-22T14:21:46.120933Z"Voigt, Christian"https://zbmath.org/authors/?q=ai:voigt.christianSummary: We define and study quantum permutations of infinite sets. This leads to discrete quantum groups which can be viewed as infinite variants of the quantum permutation groups introduced by \textit{S.-Z. Wang} [Commun. Math. Phys. 195, No.~1, 195--211 (1998; Zbl 1013.17008)]. More precisely, the resulting quantum groups encode universal quantum symmetries of the underlying sets among all discrete quantum groups. We also discuss quantum automorphisms of infinite graphs, including some examples and open problems regarding both the existence and non-existence of quantum symmetries in this setting.Backward extensions of weighted shifts on directed treeshttps://zbmath.org/1517.470582023-09-22T14:21:46.120933Z"Pikul, Piotr"https://zbmath.org/authors/?q=ai:pikul.piotrSummary: The weighted shifts are long known and form an important class of operators. One of generalisations of this class are weighted shifts on directed trees, where the linear order of coordinates in \(\ell^2\) is replaced by a more involved graph structure. In this paper, we focus on the question of joint backward extending of a given family of weighted shifts on directed trees to a weighted shift on an enveloping directed tree that preserves subnormality or power hyponormality of considered operators. One of the main results shows that the existence of such a ``joint backward extension'' for a family of weighted shifts on directed trees depends only on the possibility of backward extending of single weighted shifts that are members of the family. We introduce a generalised framework of weighted shifts on directed forests (disjoint families of directed trees) which seems to be more convenient to work with. A~characterisation of leafless directed forests on which all hyponormal weighted shifts are power hyponormal is given.A time-dependent switching mean-field game on networks motivated by optimal visiting problemshttps://zbmath.org/1517.490232023-09-22T14:21:46.120933Z"Bagagiolo, Fabio"https://zbmath.org/authors/?q=ai:bagagiolo.fabio"Marzufero, Luciano"https://zbmath.org/authors/?q=ai:marzufero.lucianoIn this paper, the authors investigate a time-dependent switching mean-field game on networks motivated by optimal visiting problems. More precisely, the authors prove the existence of a suitable definition of an approximated \(\varepsilon\)-mean-field equilibrium and then address the passage to the limit when \(\varepsilon\) goes to 0.
Reviewer: Savin Treanţă (Bucureşti)Tight sets and \(m\)-ovoids of generalised quadrangleshttps://zbmath.org/1517.510032023-09-22T14:21:46.120933Z"Bamberg, John"https://zbmath.org/authors/?q=ai:bamberg.john"Law, Maska"https://zbmath.org/authors/?q=ai:law.maska"Penttila, Tim"https://zbmath.org/authors/?q=ai:penttila.timSummary: The concept of a \textit{tight set} of points of a generalised quadrangle was introduced by \textit{S. E. Payne} in 1987 [in: Combinatorics, graph theory, and computing. Proceedings of the 18th Southeastern conference on combinatorics, graph theory, and computing, Boca Raton, FL, USA, February 23--27, 1987. Winnipeg: Utilitas Mathematica Publishing Incorporated. University of Manitoba. 243--260 (1987; Zbl 0649.51005)], and that of an \(m\)-ovoid of a generalised quadrangle was introduced by J. A. Thas in 1989, and we unify these two concepts by defining \textit{intriguing sets} of points. We prove that every intriguing set of points in a generalised quadrangle is an \(m\)-ovoid or a tight set, and we state an intersection result concerning these objects. In the classical generalised quadrangles, we construct new \(m\)-ovoids and tight sets. In particular, we construct \(m\)-ovoids of \(W(3,q)\), \(q\) odd, for all even \(m\); we construct \((q+1)/2\)-ovoids of \(W(3,q)\) for \(q\) odd; and we give a lower bound on \(m\) for \(m\)-ovoids of \(H(4,q^2)\).On the topology of bi-cyclopermutohedrahttps://zbmath.org/1517.510102023-09-22T14:21:46.120933Z"Deshpande, Priyavrat"https://zbmath.org/authors/?q=ai:deshpande.priyavrat"Manikandan, Naageswaran"https://zbmath.org/authors/?q=ai:manikandan.naageswaran"Singh, Anurag"https://zbmath.org/authors/?q=ai:singh.anurag\textit{G. Yu. Panina} [Proc. Steklov Inst. Math. 288, No. 1, 132--144 (2015; Zbl 1322.51013)] has introduced an \((n-2)\)-dimensional regular CW complex whose \(k\)-cells are labeled by cyclically ordered partitions of \(\{1, 2, \ldots, n + 1\}]\) into \((n + 1-k)\) non-empty parts, where \((n + 1-k) > 2\) -- the boundary relations in the complex corresponding to the orientation preserving refinement of partitions -- called a cyclopermutohedron and denoted by \(CP_{n+1}\). Using discrete Morse theory, \textit{I. Nekrasov} et al. [Eur. J. Math. 2 , no. 3, 835--852 (2016; Zbl 1361.51015)] showed that the homology groups \(H_i(CP_{n+1})\) are torsion free for all \(i\geq 0\) and computed their Betti numbers.
\(CP_{n+1}\) admits a free \({\mathbb Z}_2\) action; the quotient space \(CP_{n+1}/{\mathbb Z}_2\) is called a bi-cyclopermutohedron and denoted by \(QP_{n+1}\). The aim of this paper is to compute the \({\mathbb Z}_2\)- and the \({\mathbb Z}\)-homology groups of \(QP_{n+1}\).
Reviewer: Victor V. Pambuccian (Glendale)Enumeration of corner polyhedra and 3-connected Schnyder labelingshttps://zbmath.org/1517.510112023-09-22T14:21:46.120933Z"Fusy, Éric"https://zbmath.org/authors/?q=ai:fusy.eric"Narmanli, Erkan"https://zbmath.org/authors/?q=ai:narmanli.erkan"Schaeffer, Gilles"https://zbmath.org/authors/?q=ai:schaeffer.gillesSummary: We show that corner polyhedra and 3-connected Schnyder labelings join the growing list of planar structures that can be set in exact correspondence with (weighted) models of quadrant walks via a bijection due to \textit{R. Kenyon} et al. [Ann. Probab. 47, No. 3, 1240--1269 (2019; Zbl 1466.60170)].
Our approach leads to a first polynomial time algorithm to count these structures, and to the determination of their exact asymptotic growth constants: the number \(p_n\) of corner polyhedra and \(s_n\) of 3-connected Schnyder woods of size \(n\) respectively satisfy \((p_n)^{1/n}\to 9/2\) and \((s_n)^{1/n}\to 16/3\) as \(n\) goes to infinity.
While the growth rates are rational, like in the case of previously known instances of such correspondences, the exponent of the asymptotic polynomial correction to the exponential growth does not appear to follow from the now standard Denisov-Wachtel approach, due to a bimodal behavior of the step set of the underlying tandem walk. However a heuristic argument suggests that these exponents are \(-1-\pi/\arccos(9/16)\approx -4.23\) for \(p_n\) and \(-1-\pi/\arccos(22/27)\approx -6.08\) for \(s_n\), which would imply that the associated series are not D-finite.Maniplexes with automorphism group \(\mathrm{PSL}(2,q)\)https://zbmath.org/1517.520082023-09-22T14:21:46.120933Z"Leemans, Dimitri"https://zbmath.org/authors/?q=ai:leemans.dimitri"Toledo, Micael"https://zbmath.org/authors/?q=ai:toledo.micaelThis is a study of the automorphism groups of regular maniplexes. A maniplex of rank \(n\) is a combinatorial object that generalises the notion of a rank \(n\) abstract polytope. A maniplex is said to be regular if it has the highest possible degree of symmetry. The main results are: (i) there is a rank \(4\) regular maniplex with automorphism group \(\mathrm{PSL}_2(q)\) for infinitely many prime powers \(q\), (ii) no regular maniplex of rank \(n > 4\) exists that has \(\mathrm{PSL}_2(q)\) as its full automorphism group, and (iii) there are no regular maniplexes of rank \(4\) with automorphism group \(\mathrm{PSL}_2(2^k)\).
Reviewer: Victor V. Pambuccian (Glendale)The periodic steady-state solution for queues with Erlang arrivals and service and time-varying periodic transition rateshttps://zbmath.org/1517.601192023-09-22T14:21:46.120933Z"Margolius, B. H."https://zbmath.org/authors/?q=ai:margolius.barbara-hSummary: We study a queueing system with Erlang arrivals with \(k\) phases and Erlang service with \(m\) phases. Transition rates among phases vary periodically with time. For these systems, we derive an analytic solution for the asymptotic periodic distribution of the level and phase as a function of time within the period. The asymptotic periodic distribution is analogous to a steady-state distribution for a system with constant rates. If the time within the period is considered part of the state, then it is a steady-state distribution. We also obtain waiting time and busy period distributions. These solutions are expressed as infinite series. We provide bounds for the error of the estimate obtained by truncating the series. Examples are provided comparing the solution of the system of ordinary differential equation with a truncated state space to these asymptotic solutions involving remarkably few terms of the infinite series. The method can be generalized to other level independent quasi-birth-death processes if the singularities of the generating function are known.Improved baselines for causal structure learning on interventional datahttps://zbmath.org/1517.620442023-09-22T14:21:46.120933Z"Richter, Robin"https://zbmath.org/authors/?q=ai:richter.robin"Bhamidi, Shankar"https://zbmath.org/authors/?q=ai:bhamidi.shankar"Mukherjee, Sach"https://zbmath.org/authors/?q=ai:mukherjee.sachSummary: Causal structure learning (CSL) refers to the estimation of causal graphs from data. Causal versions of tools such as ROC curves play a prominent role in empirical assessment of CSL methods and performance is often compared with ``random'' baselines (such as the diagonal in an ROC analysis). However, such baselines do not take account of constraints arising from the graph context and hence may represent a ``low bar''. In this paper, motivated by examples in systems biology, we focus on assessment of CSL methods for multivariate data where part of the graph structure is known via interventional experiments. For this setting, we put forward a new class of baselines called graph-based predictors (GBPs). In contrast to the ``random'' baseline, GBPs leverage the known graph structure, exploiting simple graph properties to provide improved baselines against which to compare CSL methods. We discuss GBPs in general and provide a detailed study in the context of transitively closed graphs, introducing two conceptually simple baselines for this setting, the observed in-degree predictor (OIP) and the transitivity assuming predictor (TAP). While the former is straightforward to compute, for the latter we propose several simulation strategies. Moreover, we study and compare the proposed predictors theoretically, including a result showing that the OIP outperforms in expectation the ``random'' baseline on a subclass of latent network models featuring positive correlation among edge probabilities. Using both simulated and real biological data, we show that the proposed GBPs outperform random baselines in practice, often substantially. Some GBPs even outperform standard CSL methods (whilst being computationally cheap in practice). Our results provide a new way to assess CSL methods for interventional data.On the number of contingency tables and the independence heuristichttps://zbmath.org/1517.620672023-09-22T14:21:46.120933Z"Lyu, Hanbaek"https://zbmath.org/authors/?q=ai:lyu.hanbaek"Pak, Igor"https://zbmath.org/authors/?q=ai:pak.igorSummary: We obtain sharp asymptotic estimates on the number of \(n \times n\) contingency tables with two linear margins \(Cn\) and \(BCn\). The results imply a second-order phase transition on the number of such contingency tables, with a critical value at \(B_c:=1 + \sqrt{1+1/C} \). As a consequence, for \(B>B_c\), we prove that the classical \textit{independence heuristic} leads to a large undercounting.Component conditional fault tolerance of hierarchical folded cubic networkshttps://zbmath.org/1517.680642023-09-22T14:21:46.120933Z"Sun, Xueli"https://zbmath.org/authors/?q=ai:sun.xueli"Fan, Jianxi"https://zbmath.org/authors/?q=ai:fan.jianxi"Cheng, Baolei"https://zbmath.org/authors/?q=ai:cheng.baolei"Liu, Zhao"https://zbmath.org/authors/?q=ai:liu.zhao.1|liu.zhao"Yu, Jia"https://zbmath.org/authors/?q=ai:yu.jiaSummary: For the sake of achieving higher reliability, conditional connectivity has gradually become well-known. Component connectivity, as a kind of conditional connectivity, is an extension of the classical connectivity. Given a simple and undirected graph \(G\) and a nonnegative integer \(r\), the \((r + 1)\)-component connectivity of graph \(G\), say \(c \kappa_{r + 1}(G)\), is the minimum number of a vertex cut whose removal causes the surviving graph to have at least \(r + 1\) components. Another basic parameter of reliability, diagnosability is often related to the number of components in the surviving graph. The \((r + 1)\)-component diagnosability, say \(c t_{r + 1}(G)\), is the maximum size of fault sets subject to at least \(r + 1\) components in the surviving graph, provided that all faulty vertices can be detected. A graph is \(t / k\)-diagnosable if all faulty vertices can be isolated into a set in which up to \(k\) vertices are fault-free, provided that the number of faulty vertices is at most \(t\). As a two-level interconnection network, the hierarchical folded cubic network \(H F Q(n)\) possesses a great deal of nice properties. In this paper, we first show that the \(n\)-dimensional hierarchical folded cubic network is tightly super connected. Then, we explore that \(c \kappa_{r + 1}(H F Q(n)) = r(n + 1) - \binom {r} {2} + 1\) (\(n \geq 4\), \(1 \leq r \leq n - 4\)), and obtain that \(c t_{r + 1}(H F Q(n)) = (r + 1) n - \binom {r} {2} + 2\) (\(n \geq 4\), \(1 \leq r \leq n - 4\)) under the PMC and MM* models. Last, we show that the hierarchical folded cubic network \(H F Q(n)\) is \([(k + 1) n - \binom {k} {2} + 2] / k\)-diagnosable, where \(n \geq 4\) and \(1 \leq k \leq n - 4\).Privacy-preserving data splitting: a combinatorial approachhttps://zbmath.org/1517.681092023-09-22T14:21:46.120933Z"Farràs, Oriol"https://zbmath.org/authors/?q=ai:farras.oriol"Ribes-González, Jordi"https://zbmath.org/authors/?q=ai:ribes-gonzalez.jordi"Ricci, Sara"https://zbmath.org/authors/?q=ai:ricci.saraSummary: Privacy-preserving data splitting is a technique that aims to protect data privacy by storing different fragments of data in different locations. In this work we give a new combinatorial formulation to the data splitting problem. We see the data splitting problem as a purely combinatorial problem, in which we have to split data attributes into different fragments in a way that satisfies certain combinatorial properties derived from processing and privacy constraints. Using this formulation, we develop new combinatorial and algebraic techniques to obtain solutions to the data splitting problem. We present an algebraic method which builds an optimal data splitting solution by using Gröbner bases. Since this method is not efficient in general, we also develop a greedy algorithm for finding solutions that are not necessarily minimally sized.Deque languages, automata and planar graphshttps://zbmath.org/1517.681812023-09-22T14:21:46.120933Z"Crespi Reghizzi, Stefano"https://zbmath.org/authors/?q=ai:crespi-reghizzi.stefano"San Pietro, Pierluigi"https://zbmath.org/authors/?q=ai:san-pietro.pierluigi-lSummary: The memory of a deque automaton is more general than a queue or two stacks; to avoid overgeneralization, we consider quasi-real-time operation. Normal forms of such automata are given. Deque languages form an AFL but not a full one. We define the characteristic deque language, CDL, which combines Dyck and AntiDyck (or FIFO) languages, and homomorphically characterizes the deque languages. The notion of deque graph, from graph theory, well represents deque computation by means of a planar Hamiltonian graph on a cylinder, with edges visualizing producer-consumer relations for deque symbols. We give equivalent definitions of CDL by labelled deque graphs, by cancellation rules, and by means of shuffle and intersection of simpler languages. The labeled deque graph of a sentence generalizes traditional syntax trees. The layout of deque computations on a cylinder is remindful of 3D models used in theoretical (bio)chemistry.
For the entire collection see [Zbl 1398.68030].Enumerating regular expressions and their languageshttps://zbmath.org/1517.681902023-09-22T14:21:46.120933Z"Gruber, Hermann"https://zbmath.org/authors/?q=ai:gruber.hermann"Lee, Jonathan"https://zbmath.org/authors/?q=ai:lee.jonathan-k|lee.jonathan-d|lee.jonathan-t|lee.jonathan-w"Shallit, Jeffrey"https://zbmath.org/authors/?q=ai:shallit.jeffrey-oThis chapter of the Handbook guides the reader through the process of enumerating rational expressions of a given length and, subsequently, obtaining lower and upper bounds for the number of rational languages determined by rational expressions of given length. The authors consider three different notions of length in this context and summarise the basic relations among them.
The enumeration of rational expressions of given length can be performed using the standard methodology of \textit{analytic combinatorics} -- see, e.g., [\textit{P. Flajolet} and \textit{R. Sedgewick}, Analytic combinatorics. Cambridge: Cambridge University Press (2009; Zbl 1165.05001)]. The key is to observe that the language of all rational expressions can be generated by an unambiguous context-free grammar. This grammar may differ depending on precisely which rational expressions one considers to be valid (for instance, it is debatable whether one should allow the empty expression or expressions containing superfluous parentheses). The authors explicitly describe an unambiguous context-free grammar that they use as a specification of valid rational expressions in their considerations.
Given an unambiguous context-free grammar generating some subset \(S\) of all rational expressions, it follows by the Chomsky-Schützenberger enumeration theorem that if \(a_n\) denotes the number of rational expressions of (ordinary) length \(n\) in \(S\), then the generating function of the sequence of all such \(a_n\) is algebraic. In fact, the grammar can be directly transformed to an \(\mathbb{N}\)-algebraic system of equations such that the said generating function is the first component of its unique solution. Using Gröbner bases, one can further transform this system to a single algebraic equation satisfied by the generating function. This process is informally explained by the authors.
An exact asymptotic estimate for \(a_n\) as \(n\) tends to infinity can be obtained from such an algebraic equation via singularity analysis. Nevertheless, this is not strictly necessary for the enumeration of rational languages as presented by the authors, so the reader is only referred to [loc. cit.] regarding these matters. Instead, the authors describe in detail how to obtain simple lower and upper bounds for \(a_n\) by making use of Pringsheim's theorem.
Lower bounds for the number of rational languages determined by rational expressions of given length are then obtained by applying these observations to unambiguous grammars for certain particular families of rational expressions such that any two distinct expressions in this family determine different rational languages. Any lower bound for the number of such expressions thus also gives a lower bound for the number of rational languages. On the other hand, upper bounds are obtained by considering certain normal forms of rational expressions, i.e., families of expressions that still can be used to determine every rational language. Any upper bound for the number of such expressions yields an upper bound for the number of rational languages.
For the entire collection see [Zbl 1470.68001].
Reviewer: Peter Kostolányi (Bratislava)Dynamics of the independence number and automata synchronizationhttps://zbmath.org/1517.681932023-09-22T14:21:46.120933Z"Gusev, Vladimir V."https://zbmath.org/authors/?q=ai:gusev.vladimir-v"Jungers, Raphaël M."https://zbmath.org/authors/?q=ai:jungers.raphael-m"Průša, Daniel"https://zbmath.org/authors/?q=ai:prusa.danielSummary: We study the lengths of synchronizing words produced by the classical greedy compression algorithm. We associate a sequence of graphs with every synchronizing automaton and rely on evolution of the independence number to bound the lengths of produced words. By leveraging graph theoretical results we show that in many cases automata with good extension properties have good compression properties as well. More precisely, we show that the compression algorithm will produce a synchronizing word of length \(\mathcal{O}(n^2\log (n))\) on cyclic, regular and strongly-transitive automata with \(n\) states, which is not far from the best possible bound of \((n-1)^2\). Furthermore, we provide a relatively simple proof that every \(n\)-state automaton has a synchronizing word of length at most \(\frac{n^3}{4}+\mathcal{O}(n^2)\).
For the entire collection see [Zbl 1398.68030].Exhaustive generation of some lattice paths and their prefixeshttps://zbmath.org/1517.682642023-09-22T14:21:46.120933Z"Barcucci, Elena"https://zbmath.org/authors/?q=ai:barcucci.elena"Bernini, Antonio"https://zbmath.org/authors/?q=ai:bernini.antonio"Pinzani, Renzo"https://zbmath.org/authors/?q=ai:pinzani.renzoSummary: We refer to positive lattice paths as to paths in the discrete plane constituted by different kinds of steps (north-east, east and south-east), starting from the origin and never going under the \(x\)-axis. They have been deeply studied both from a combinatorial and an algorithmic point of view. We propose some algorithms for the exhaustive generation of positive paths which are Motzkin and Schröder paths and their prefixes, according to their length. For each kind of paths we define a recursive algorithm as well as an iterative one, specifying which path follows a given one in the lexicographic order. Furthermore we study the complexity of these algorithms by using the relations between the number of paths of a given size and the number of final north-east or south-east steps.New and improved algorithms for unordered tree inclusionhttps://zbmath.org/1517.682672023-09-22T14:21:46.120933Z"Akutsu, Tatsuya"https://zbmath.org/authors/?q=ai:akutsu.tatsuya"Jansson, Jesper"https://zbmath.org/authors/?q=ai:jansson.jesper"Li, Ruiming"https://zbmath.org/authors/?q=ai:li.ruiming"Takasu, Atsuhiro"https://zbmath.org/authors/?q=ai:takasu.atsuhiro"Tamura, Takeyuki"https://zbmath.org/authors/?q=ai:tamura.takeyukiSummary: The \textit{tree inclusion problem} is, given two node-labeled trees \(P\) and \(T\) (the ``pattern tree'' and the ``target tree''), to locate every minimal subtree in \(T\) (if any) that can be obtained by applying a sequence of node insertion operations to \(P\). Although the \textit{ordered} tree inclusion problem is solvable in polynomial time, the \textit{unordered} tree inclusion problem is NP-hard. The currently fastest algorithm for the latter is a classic algorithm by Kilpeläinen and Mannila from 1995 that runs in \(O(d 2^{2 d} m n)\) time, where \(m\) and \(n\) are the sizes of the pattern and target trees, respectively, and \(d\) is the degree of the pattern tree. Here, we develop a new algorithm that runs in \(O(d 2^d m n^2)\) time, improving the exponential factor from \(2^{2 d}\) to \(2^d\) by considering a particular type of ancestor-descendant relationships that is suitable for dynamic programming. We also study restricted variants of the unordered tree inclusion problem.New and improved algorithms for unordered tree inclusionhttps://zbmath.org/1517.682682023-09-22T14:21:46.120933Z"Akutsu, Tatsuya"https://zbmath.org/authors/?q=ai:akutsu.tatsuya"Jansson, Jesper"https://zbmath.org/authors/?q=ai:jansson.jesper"Li, Ruiming"https://zbmath.org/authors/?q=ai:li.ruiming"Takasu, Atsuhiro"https://zbmath.org/authors/?q=ai:takasu.atsuhiro"Tamura, Takeyuki"https://zbmath.org/authors/?q=ai:tamura.takeyukiSummary: The tree inclusion problem is, given two node-labeled trees \(P\) and \(T\) (the ``pattern tree'' and the ``text tree''), to locate every minimal subtree in \(T\) (if any) that can be obtained by applying a sequence of node insertion operations to \(P\). Although the ordered tree inclusion problem is solvable in polynomial time, the unordered tree inclusion problem is NP-hard. The currently fastest algorithm for the latter is from 1995 and runs in \(O(\operatorname{poly}(m,n)\cdot 2^{2d})= O^*(2^{2d})\) time, where \(m\) and \(n\) are the sizes of the pattern and text trees, respectively, and \(d\) is the maximum outdegree of the pattern tree. Here, we develop a new algorithm that improves the exponent \(2d\) to \(d\) by considering a particular type of ancestor-descendant relationships and applying dynamic programming, thus reducing the time complexity to \(O^*(2^d)\). We then study restricted variants of the unordered tree inclusion problem where the number of occurrences of different node labels and/or the input trees' heights are bounded. We show that although the problem remains NP-hard in many such cases, it can be solved in polynomial time for \(c=2\) and in \(O^*(1.8^d)\) time for \(c=3\) if the leaves of \(P\) are distinctly labeled and each label occurs at most \(c\) times in \(T\). We also present a randomized \(O^*(1.883^d)\)-time algorithm for the case that the heights of \(P\) and \(T\) are one and two, respectively.
For the entire collection see [Zbl 1407.68036].Recoloring the colored de Bruijn graphhttps://zbmath.org/1517.682692023-09-22T14:21:46.120933Z"Alipanahi, Bahar"https://zbmath.org/authors/?q=ai:alipanahi.bahar"Kuhnle, Alan"https://zbmath.org/authors/?q=ai:kuhnle.alan"Boucher, Christina"https://zbmath.org/authors/?q=ai:boucher.christinaSummary: The colored de Bruijn graph, an extension of the de Bruijn graph, is routinely applied for variant calling, genotyping, genome assembly, and various other applications. In this data structure, the edges are labeled with one or more colors from a set \(\{c_1, \dots , c_{\alpha } \}\), and are stored as a \(m \times \alpha\) matrix, where \(m\) is the number of edges. Recently, there has been a significant amount of work in developing compacted representations of this color matrix but all existing methods have focused on compressing the color matrix. In this paper, we explore the problem of recoloring the graph in order to reduce the number of colors, and thus, decrease the size of the color matrix. We show that finding the minimum number of colors needed for recoloring is not only NP-hard but also, difficult to approximate within a reasonable factor. These hardness results motivate the need for a recoloring heuristic that we present in this paper. Our results show that this heuristic is able to reduce the number of colors between one and two orders of magnitude. More specifically, when the number of colors is large (>5,000,000) the number of colors is reduced by a factor of 136 by our heuristic. An implementation of this heuristic is publicly available at \url{https://github.com/baharpan/cosmo/tree/Recoloring}.
For the entire collection see [Zbl 1398.68028].Finding cores of limited lengthhttps://zbmath.org/1517.682702023-09-22T14:21:46.120933Z"Alstrup, Stephen"https://zbmath.org/authors/?q=ai:alstrup.stephen"Lauridsen, Peter W."https://zbmath.org/authors/?q=ai:lauridsen.peter-w"Sommerlund, Peer"https://zbmath.org/authors/?q=ai:sommerlund.peer"Thorup, Mikkel"https://zbmath.org/authors/?q=ai:thorup.mikkelSummary: In this paper we consider the problem of finding a core of limited length in a tree. A core is a path, which minimizes the sum of the distances to all nodes in the tree. This problem has been examined under different constraints on the tree and on the set of paths, from which the core can be chosen. For all cases, we present linear or almost linear time algorithms, which improves the previous results due to
[\textit{S. Peng} and \textit{W.-t. Lo}, J. Algorithms 20, No. 3, 445--458 (1996; Zbl 0845.68053); \textit{E. Minieka}, Networks 15, 309--321 (1985; Zbl 0579.90027)].
For the entire collection see [Zbl 1492.68014].On minimum connecting transition sets in graphshttps://zbmath.org/1517.682722023-09-22T14:21:46.120933Z"Bellitto, Thomas"https://zbmath.org/authors/?q=ai:bellitto.thomas"Bergougnoux, Benjamin"https://zbmath.org/authors/?q=ai:bergougnoux.benjaminSummary: A forbidden transition graph is a graph defined together with a set of permitted transitions i.e. unordered pair of adjacent edges that one may use consecutively in a walk in the graph. In this paper, we look for the smallest set of transitions needed to be able to go from any vertex of the given graph to any other. We prove that this problem is NP-hard and study approximation algorithms. We develop theoretical tools that help to study this problem.
For the entire collection see [Zbl 1398.68016].Token sliding on split graphshttps://zbmath.org/1517.682732023-09-22T14:21:46.120933Z"Belmonte, Rémy"https://zbmath.org/authors/?q=ai:belmonte.remy"Kim, Eun Jung"https://zbmath.org/authors/?q=ai:kim.eun-jung"Lampis, Michael"https://zbmath.org/authors/?q=ai:lampis.michael"Mitsou, Valia"https://zbmath.org/authors/?q=ai:mitsou.valia"Otachi, Yota"https://zbmath.org/authors/?q=ai:otachi.yota"Sikora, Florian"https://zbmath.org/authors/?q=ai:sikora.florianSummary: We consider the complexity of the \textsc{Independent Set Reconfiguration} problem under the Token Sliding rule. In this problem we are given two independent sets of a graph and are asked if we can transform one to the other by repeatedly exchanging a vertex that is currently in the set with one of its neighbors, while maintaining the set independent. Our main result is to show that this problem is PSPACE-complete on split graphs (and hence also on chordal graphs), thus resolving an open problem in this area. We then go on to consider the \(c\)-\textsc{Colorable Reconfiguration} problem under the same rule, where the constraint is now to maintain the set \(c\)-colorable at all times. As one may expect, a simple modification of our reduction shows that this more general problem is PSPACE-complete for all fixed \(c\geq 1\) on chordal graphs. Somewhat surprisingly, we show that the same cannot be said for split graphs: we give a polynomial time \((n^{O(c)})\) algorithm for all fixed values of \(c\), except \(c=1\), for which the problem is PSPACE-complete. We complement our algorithm with a lower bound showing that \(c\)-\textsc{Colorable Reconfiguration} is W[2]-hard on split graphs parameterized by \(c\) and the length of the solution, as well as a tight ETH-based lower bound for both parameters. Finally, we study \(c\)-\textsc{Colorable Reconfiguration} under a relaxed rule called Token Jumping, where exchanged vertices are not required to be adjacent. We show that the problem on chordal graphs is PSPACE-complete even if \(c\) is some fixed constant. We then show that the problem is polynomial-time solvable for strongly chordal graphs even if \(c\) is a part of the input.Token sliding on split graphshttps://zbmath.org/1517.682742023-09-22T14:21:46.120933Z"Belmonte, Rémy"https://zbmath.org/authors/?q=ai:belmonte.remy"Kim, Eun Jung"https://zbmath.org/authors/?q=ai:kim.eun-jung"Lampis, Michael"https://zbmath.org/authors/?q=ai:lampis.michael"Mitsou, Valia"https://zbmath.org/authors/?q=ai:mitsou.valia"Otachi, Yota"https://zbmath.org/authors/?q=ai:otachi.yota"Sikora, Florian"https://zbmath.org/authors/?q=ai:sikora.florianSummary: We consider the complexity of the Independent Set Reconfiguration problem under the Token Sliding rule. In this problem we are given two independent sets of a graph and are asked if we can transform one to the other by repeatedly exchanging a vertex that is currently in the set with one of its neighbors, while maintaining the set independent. Our main result is to show that this problem is PSPACE-complete on split graphs (and hence also on chordal graphs), thus resolving an open problem in this area.\par We then go on to consider the \(c\)-Colorable Reconfiguration problem under the same rule, where the constraint is now to maintain the set \(c\)-colorable at all times. As one may expect, a simple modification of our reduction shows that this more general problem is PSPACE-complete for all fixed \(c\ge 1\) on chordal graphs. Somewhat surprisingly, we show that the same cannot be said for split graphs: we give a polynomial time \((n^{O(c)})\) algorithm for all fixed values of \(c\), except \(c=1\), for which the problem is PSPACE-complete. We complement our algorithm with a lower bound showing that \(c\)-Colorable Reconfiguration is W[2]-hard on split graphs parameterized by \(c\) and the length of the solution, as well as a tight ETH-based lower bound for both parameters.
For the entire collection see [Zbl 1411.68018].A deterministic distributed 2-approximation for weighted vertex cover in \(O(\log N\log\varDelta/\log^2\log\varDelta)\) roundshttps://zbmath.org/1517.682752023-09-22T14:21:46.120933Z"Ben-Basat, Ran"https://zbmath.org/authors/?q=ai:ben-basat.ran"Even, Guy"https://zbmath.org/authors/?q=ai:even.guy"Kawarabayashi, Ken-ichi"https://zbmath.org/authors/?q=ai:kawarabayashi.ken-ichi"Schwartzman, Gregory"https://zbmath.org/authors/?q=ai:schwartzman.gregorySummary: We present a deterministic distributed 2-approximation algorithm for the Minimum Weight Vertex Cover problem in the CONGEST model whose round complexity is \(O(\log n\log\varDelta/\log ^2\log\varDelta)\). This improves over the currently best known deterministic 2-approximation implied by [\textit{S. Khuller} et al., J. Algorithms 17, No. 2, 280--289 (1994; Zbl 0938.68946)]. Our solution generalizes the \((2+\epsilon)\)-approximation algorithm of [\textit{R. Bar-Yehuda} et al., J. ACM 64, No. 3, Article No. 23, 11 p. (2017; Zbl 1426.68291)], improving the dependency on \(\epsilon^{-1}\) from linear to logarithmic. In addition, for every \(\epsilon=(\log\varDelta)^{-c}\), where \(c\geq 1\) is a constant, our algorithm computes a \(\left(2+\epsilon\right)\)-approximation in \(O(\log{\varDelta}/\log\log{\varDelta})\) rounds (which is asymptotically optimal).
For the entire collection see [Zbl 1400.68027].Recognizing hyperelliptic graphs in polynomial timehttps://zbmath.org/1517.682762023-09-22T14:21:46.120933Z"Bodewes, Jelco M."https://zbmath.org/authors/?q=ai:bodewes.jelco-m"Bodlaender, Hans L."https://zbmath.org/authors/?q=ai:bodlaender.hans-l"Cornelissen, Gunther"https://zbmath.org/authors/?q=ai:cornelissen.gunther"van der Wegen, Marieke"https://zbmath.org/authors/?q=ai:van-der-wegen.mariekeSummary: Based on analogies between algebraic curves and graphs, Baker and Norine introduced divisorial gonality, a graph parameter for multigraphs related to treewidth, multigraph algorithms and number theory. We consider so-called hyperelliptic graphs (multigraphs of gonality 2) and provide a safe and complete set of reduction rules for such multigraphs, showing that we can recognize hyperelliptic graphs in time \(O(n\log n+m)\), where \(n\) is the number of vertices and \(m\) the number of edges of the multigraph. A corollary is that we can decide with the same runtime whether a two-edge-connected graph \(G\) admits an involution \(\sigma\) such that the quotient \(G/\langle\sigma\rangle\) is a tree.
For the entire collection see [Zbl 1398.68016].On directed feedback vertex set parameterized by treewidthhttps://zbmath.org/1517.682772023-09-22T14:21:46.120933Z"Bonamy, Marthe"https://zbmath.org/authors/?q=ai:bonamy.marthe"Kowalik, Łukasz"https://zbmath.org/authors/?q=ai:kowalik.lukasz"Nederlof, Jesper"https://zbmath.org/authors/?q=ai:nederlof.jesper"Pilipczuk, Michał"https://zbmath.org/authors/?q=ai:pilipczuk.michal"Socała, Arkadiusz"https://zbmath.org/authors/?q=ai:socala.arkadiusz"Wrochna, Marcin"https://zbmath.org/authors/?q=ai:wrochna.marcinSummary: We study the Directed Feedback Vertex Set problem parameterized by the treewidth of the input graph. We prove that unless the Exponential Time Hypothesis fails, the problem cannot be solved in time \(2^{o(t\log t)}\cdot n^{\mathcal{O}(1)}\) on general directed graphs, where \(t\) is the treewidth of the underlying undirected graph. This is matched by a dynamic programming algorithm with running time \(2^{\mathcal{O}(t\log t)}\cdot n^{\mathcal {O}(1)}\). On the other hand, we show that if the input digraph is planar, then the running time can be improved to \(2^{\mathcal{O}(t)}\cdot n^{\mathcal{O}(1)}\).
For the entire collection see [Zbl 1398.68016].Optimality program in segment and string graphshttps://zbmath.org/1517.682782023-09-22T14:21:46.120933Z"Bonnet, Édouard"https://zbmath.org/authors/?q=ai:bonnet.edouard"Rzążewski, Paweł"https://zbmath.org/authors/?q=ai:rzazewski.pawelSummary: Planar graphs are known to allow subexponential algorithms running in time \(2^{O(\sqrt{n})}\) or \(2^{O(\sqrt{n}\log n)}\) for most of the paradigmatic problems, while the brute-force time \(2^{\varTheta(n)}\) is very likely to be asymptotically best on general graphs. Intrigued by an algorithm packing curves in \(2^{O(n^{2/3}\log n)}\) by \textit{J. Fox} and \textit{J. Pach} [SODA 2011, 1161--1165 (2011; Zbl 1377.68275)], we investigate which problems have subexponential algorithms on the intersection graphs of curves (string graphs) or segments (segment intersection graphs) and which problems have no such algorithms under the ETH (Exponential Time Hypothesis). Among our results, we show that, quite surprisingly, 3-Coloring can also be solved in time \(2^{O(n^{2/3}\log ^{O(1)}n)}\) on string graphs while an algorithm running in time \(2^{o(n)}\) for 4-Coloring even on axis-parallel segments (of unbounded length) would disprove the ETH. For 4-Coloring of unit segments, we show a weaker lower bound, excluding a \(2^{o(n^{2/3})}\) algorithm (under the ETH). The construction exploits the celebrated Erdős-Szekeres theorem. The subexponential running time also carries over to Min Feedback Vertex Set, but not to Min Dominating Set and Min Independent Dominating Set.
For the entire collection see [Zbl 1398.68016].Simple and local independent set approximationhttps://zbmath.org/1517.682792023-09-22T14:21:46.120933Z"Boppana, Ravi B."https://zbmath.org/authors/?q=ai:boppana.ravi-b"Halldórsson, Magnús M."https://zbmath.org/authors/?q=ai:halldorsson.magnus-m"Rawitz, Dror"https://zbmath.org/authors/?q=ai:rawitz.drorSummary: We bound the performance guarantees that follow from Turán-like bounds for unweighted and weighted independent sets in bounded-degree graphs. In particular, a randomized approach of Boppana forms a simple 1-round distributed algorithm, as well as a streaming and preemptive online algorithm. We show it gives a tight \((\varDelta+1)/2\)-approximation in unweighted graphs of maximum degree \(\varDelta\), which is best possible for 1-round distributed algorithms. For weighted graphs, it gives only a \((\varDelta+1)\)-approximation, but a simple modification results in an asymptotic expected \(0.529(\varDelta+1)\)-approximation. This compares with a recent, more complex \(\varDelta\)-approximation [\textit{R. Bar-Yehuda} et al., PODC 2017, 165--174 (2017; Zbl 1380.68416)], which holds deterministically.
For the entire collection see [Zbl 1400.68027].Temporal cliques admit sparse spannershttps://zbmath.org/1517.682802023-09-22T14:21:46.120933Z"Casteigts, Arnaud"https://zbmath.org/authors/?q=ai:casteigts.arnaud"Peters, Joseph G."https://zbmath.org/authors/?q=ai:peters.joseph-g"Schoeters, Jason"https://zbmath.org/authors/?q=ai:schoeters.jasonSummary: Let \(G=(V,E)\) be an undirected graph on \(n\) vertices and \(\lambda:E\to 2^{\mathbb{N}}\) a mapping that assigns to every edge a non-empty set of integer labels (discrete times when the edge is present). Such a labelled graph \(\mathcal{G}=(G,\lambda)\) is \textit{temporally connected} if a path exists with non-decreasing times from every vertex to every other vertex. In a seminal paper, \textit{D. Kempe} et al. [STOC 2000, 504--513 (2000; Zbl 1296.68015)] sked whether, given such a temporally connected graph, a \textit{sparse} subset of edges always exists whose labels suffice to preserve temporal connectivity -- a \textit{temporal spanner}. \textit{K. Axiotis} and \textit{D. Fotakis} [LIPIcs -- Leibniz Int. Proc. Inform. 55, Article 149, 14 p. (2016; Zbl 1388.68299)] answered negatively by exhibiting a family of \(\Theta(n^2)\)-dense temporal graphs which admit no temporal spanner of density \(o(n^2)\). In this paper, we give the first positive answer as to the existence of \(o(n^2)\)-sparse spanners in a dense class of temporal graphs, by showing (constructively) that if \(G\) is a complete graph, then one can always find a temporal spanner with \(O(n\log n)\) edges.Temporal cliques admit sparse spannershttps://zbmath.org/1517.682812023-09-22T14:21:46.120933Z"Casteigts, Arnaud"https://zbmath.org/authors/?q=ai:casteigts.arnaud"Peters, Joseph G."https://zbmath.org/authors/?q=ai:peters.joseph-g"Schoeters, Jason"https://zbmath.org/authors/?q=ai:schoeters.jasonSummary: Let \(G=(V,E)\) be an undirected graph on \(n\) vertices and \(\lambda:E\to 2^{\mathbb{N}}\) a mapping that assigns to every edge a non-empty set of positive integer labels. These labels can be seen as discrete times when the edge is present. Such a labeled graph \(\mathcal{G}=(G,\lambda)\) is said to be temporally connected if a path exists with non-decreasing times from every vertex to every other vertex. In a seminal paper, \textit{D. Kempe} et al. [STOC 2000, 504--513 (2000; Zbl 1296.68015)] asked whether, given such a temporal graph, a sparse subset of edges can always be found whose labels suffice to preserve temporal connectivity -- a temporal spanner. \textit{K. Axiotis} and \textit{D. Fotakis} [LIPIcs -- Leibniz Int. Proc. Inform. 55, Article 149, 14 p. (2016; Zbl 1388.68299)] answered negatively by exhibiting a family of \(\Theta(n^2)\)-dense temporal graphs which admit no temporal spanner of density \(o(n^2)\). The natural question is then whether sparse temporal spanners always exist in some classes of dense graphs.\par In this paper, we answer this question affirmatively, by showing that if the underlying graph \(G\) is a complete graph, then one can always find temporal spanners of density \(O(n\log n)\). The best known result for complete graphs so far was that spanners of density \(\binom{n}{2}-\lfloor n/4\rfloor= O(n^2)\) always exist. Our result is the first positive answer as to the existence of \(o(n^2)\) sparse spanners in adversarial instances of temporal graphs since the original question by Kempe et al. [loc. cit.], focusing here on complete graphs. The proofs are constructive and directly adaptable as an algorithm.
For the entire collection see [Zbl 1414.68003].On the complexity of minimum maximal uniquely restricted matchinghttps://zbmath.org/1517.682822023-09-22T14:21:46.120933Z"Chaudhary, Juhi"https://zbmath.org/authors/?q=ai:chaudhary.juhi"Panda, B. S."https://zbmath.org/authors/?q=ai:panda.bhawani-sankarSummary: A subset \(M \subseteq E\) of edges of a graph \(G = (V, E)\) is called a \textit{matching} if no two edges of \(M\) share a common vertex. A matching \(M\) in a graph \(G\) is called a \textit{uniquely restricted matching} if, \(G [V(M)]\), the subgraph of \(G\) induced by the set of \(M\)-saturated vertices of \(G\) contains exactly one perfect matching. A uniquely restricted matching \(M\) is \textit{maximal} if \(M\) is not properly contained in any uniquely restricted matching of \(G\). Given a graph \(G\), the \textsc{Min-Max-UR Matching} problem asks to find a maximal uniquely restricted matching in \(G\) of minimum cardinality and \textsc{Decide-Min-Max-UR Matching} problem, the decision version of this problem takes a graph \(G\) and an integer \(k\) and asks whether \(G\) admits a maximal uniquely restricted matching of cardinality at most \(k\). It is known that the \textsc{Decide-Min-Max-UR Matching} problem is \(\mathsf{NP} \)-complete. In this paper, we strengthen this result by proving that the \textsc{Decide-Min-Max-UR Matching} problem remains \(\mathsf{NP} \)-complete for chordal bipartite graphs, star-convex bipartite graphs, chordal graphs, and doubly chordal graphs. On the positive side, we prove that the \textsc{Min-Max-UR Matching} problem is polynomial time solvable for bipartite distance-hereditary graphs and linear time solvable for bipartite permutation graphs, proper interval graphs, and threshold graphs. Finally, we prove that the \textsc{Min-Max-UR Matching} problem is \(\mathsf{APX} \)-complete for graphs with maximum degree 4.On the complexity of minimum maximal uniquely restricted matchinghttps://zbmath.org/1517.682832023-09-22T14:21:46.120933Z"Chaudhary, Juhi"https://zbmath.org/authors/?q=ai:chaudhary.juhi"Panda, B. S."https://zbmath.org/authors/?q=ai:panda.bhawani-sankarSummary: A subset \(M\subseteq E\) of edges of a graph \(G=(V,E)\) is called a \textit{matching} if no two edges of \(M\) share a common vertex. A matching \(M\) in a graph \(G\) is called a \textit{uniquely restricted matching} if \(G[V(M)]\), the subgraph of \(G\) induced by the \(M\)-saturated vertices of \(G\), contains exactly one perfect matching. A uniquely restricted matching \(M\) is \textit{maximal} if \(M\) is not properly contained in any other uniquely restricted matching of \(G\). Given a graph \(G\), the \textsc{Min-Max-UR Matching} problem asks to find a maximal uniquely restricted matching of minimum cardinality in \(G\). In general, the decision version of the \textsc{Min-Max-UR Matching} problem is known to be \({\mathsf{NP}} \)-complete for general graphs and remains so even for bipartite graphs. In this paper, we strengthen this result by proving that this problem remains \({\mathsf{NP}} \)-complete for chordal bipartite graphs and chordal graphs. On the positive side, we prove that the \textsc{Min-Max-UR Matching} problem is polynomial time solvable for bipartite permutation graphs and proper interval graphs. Finally, we show that the \textsc{Min-Max-UR Matching} problem is \(\mathsf{APX} \)-complete for bounded degree graphs.
For the entire collection see [Zbl 1507.68048].Constructing dual-CISTs with short diameters using a generic adjustment scheme on bicubeshttps://zbmath.org/1517.682842023-09-22T14:21:46.120933Z"Chen, Yu-Han"https://zbmath.org/authors/?q=ai:chen.yuhan"Tang, Shyue-Ming"https://zbmath.org/authors/?q=ai:tang.shyue-ming"Pai, Kung-Jui"https://zbmath.org/authors/?q=ai:pai.kung-jui"Chang, Jou-Ming"https://zbmath.org/authors/?q=ai:chang.jou-mingSummary: Recently, an innovative hypercube-variant network called bicube, denoted as \(B Q_n\), has been proposed to possess both short diameter and symmetry advantages. Unlike other existing hypercube-variant networks, they lose their symmetry in pursuit of short diameters. For solving the problems of fault-tolerant transmission and secure message distribution in a reliable network, one solution suggested a dual-CIST (two completely independent spanning trees) to design a multi-path routing (e.g., a recently proposed secure-protection routing). We can make the construction using the standard arrangement guideline (SAG) like the hypercubes to obtain a dual-CIST with a diameter of \(2 n - 1\) on \(B Q_n\). This paper proposes a newly generic adjustment scheme (GAS) for reducing the diameter of the dual-CIST under this construction. As a result, the diameter of \(T_i\) for \(i = 1, 2\) we constructed for \(B Q_n\) are as follows:
\[\operatorname{diam}(T_i) =
\begin{cases}
7 \quad & \text{if } n = 4 ; \\
2 n - 2 \quad & \text{if } n \geqslant 5 \text{ and } n \text{ is odd}; \\
2 n - 3 \quad & \text{if } n \geqslant 6 \text{ and } n \text{ is even}.
\end{cases}\]Independent sets in vertex-arrival streamshttps://zbmath.org/1517.682852023-09-22T14:21:46.120933Z"Cormode, Graham"https://zbmath.org/authors/?q=ai:cormode.graham"Dark, Jacques"https://zbmath.org/authors/?q=ai:dark.jacques"Konrad, Christian"https://zbmath.org/authors/?q=ai:konrad.christianSummary: We consider the maximal and maximum independent set problems in three models of graph streams:
\begin{itemize}\item[--] In the edge model we see a stream of edges which collectively define a graph; this model is well-studied for a variety of problems. We show that the space complexity for a one-pass streaming algorithm to find a maximal independent set is quadratic (i.e. we must store all edges). We further show that it is not much easier if we only require approximate maximality. This contrasts strongly with the other two vertex-based models, where one can greedily find an exact solution in only the space needed to store the independent set.\item[--] In the ``explicit'' vertex model, the input stream is a sequence of vertices making up the graph. Every vertex arrives along with its incident edges that connect to previously arrived vertices. Various graph problems require substantially less space to solve in this setting than in edge-arrival streams. We show that every one-pass \(c\)-approximation streaming algorithm for maximum independent set (MIS) on explicit vertex streams requires \(\Omega(\frac{n^2}{c^6})\) bits of space, where \(n\) is the number of vertices of the input graph. It is already known that \(\widetilde{\Theta}(\frac{n^2}{c^2})\) bits of space are necessary and sufficient in the edge arrival model [\textit{M. M. Halldórsson} et al., Lect. Notes Comput. Sci. 7391, 449--460 (2012; Zbl 1272.68333)], thus the MIS problem is not significantly easier to solve under the explicit vertex arrival order assumption. Our result is proved via a reduction from a new multi-party communication problem closely related to pointer jumping.\item[--] In the ``implicit'' vertex model, the input stream consists of a sequence of objects, one per vertex. The algorithm is equipped with a function that maps pairs of objects to the presence or absence of edges, thus defining the graph. This model captures, for example, geometric intersection graphs such as unit disc graphs. Our final set of results consists of several improved upper and lower bounds for interval and square intersection graphs, in both explicit and implicit streams. In particular, we show a gap between the hardness of the explicit and implicit vertex models for interval graphs.\end{itemize}
For the entire collection see [Zbl 1414.68003].Computing small pivot-minorshttps://zbmath.org/1517.682862023-09-22T14:21:46.120933Z"Dabrowski, Konrad K."https://zbmath.org/authors/?q=ai:dabrowski.konrad-kazimierz"Dross, François"https://zbmath.org/authors/?q=ai:dross.francois"Jeong, Jisu"https://zbmath.org/authors/?q=ai:jeong.jisu"Kanté, Mamadou Moustapha"https://zbmath.org/authors/?q=ai:kante.mamadou-moustapha"Kwon, O-joung"https://zbmath.org/authors/?q=ai:kwon.ojoung"Oum, Sang-il"https://zbmath.org/authors/?q=ai:oum.sang-il"Paulusma, Daniël"https://zbmath.org/authors/?q=ai:paulusma.danielSummary: A graph \(G\) contains a graph \(H\) as a pivot-minor if \(H\) can be obtained from \(G\) by applying a sequence of vertex deletions and edge pivots. Pivot-minors play an important role in the study of rank-width. However, so far, pivot-minors have only been studied from a structural perspective. We initiate a systematic study into their complexity aspects. We first prove that the Pivot-Minor problem, which asks if a given graph \(G\) contains a given graph \(H\) as a pivot-minor, is NP-complete. If \(H\) is not part of the input, we denote the problem by \(H\)-Pivot-Minor. We give a certifying polynomial-time algorithm for \(H\)-Pivot-Minor for every graph \(H\) with \(|V(H)|\leq 4\) except when \(H\in\{K_4,C_3+P_1,4P_1\}\), via a structural characterization of \(H\)-pivot-minor-free graphs in terms of a set \(\mathcal{F}_H\) of minimal forbidden induced subgraphs.
For the entire collection see [Zbl 1398.68016].Saving probe bits by cube dominationhttps://zbmath.org/1517.682882023-09-22T14:21:46.120933Z"Damaschke, Peter"https://zbmath.org/authors/?q=ai:damaschke.peterSummary: We consider the problem of storing a single element from an \(m\)-element set as a binary string of optimal length, and comparing any queried string to the stored string without reading all bits. This is the one-element version of the problem of membership testing in the bit probe model, and solutions can serve as building blocks of general membership testers. Our principal contribution is the equivalence of saving probe bits with some generalized notion of domination in hypercubes. This domination variant requires that every vertex outside the dominating set belongs to a sub-hypercube, of fixed dimension, in which all other vertices belong to in the dominating set. This fixed dimension equals the number of saved probe bits. We give specific constructions showing that up to three probe bits can be ignored when \(m\) is far enough from the next larger power of 2. The main technical idea is to use low-dimensional (grid) relaxations of the problem. The design of optimal schemes remains an open problem, however one has to notice that even usual domination in hypercubes is far from being completely understood.
For the entire collection see [Zbl 1398.68016].Graph amalgamation under logical constraintshttps://zbmath.org/1517.682892023-09-22T14:21:46.120933Z"de Oliveira Oliveira, Mateus"https://zbmath.org/authors/?q=ai:de-oliveira-oliveira.mateusSummary: We say that a graph \(G\) is an \(H\)-amalgamation of graphs \(G_1\) and \(G_2\) if \(G\) can be obtained by gluing \(G_1\) and \(G_2\) along isomorphic copies of \(H\). In the Amalgamation Recognition problem we are given connected graphs \(H,G_1,G_2,G\) and the goal is to determine whether \(G\) is an \(H\)-amalgamation of \(G_1\) and \(G_2\). Our main result states that Amalgamation Recognition can be solved in time \(2^{O(\varDelta\cdot t)}\cdot n^{O(t)}\) where \(n,t,\varDelta\) are the number of vertices, the treewidth and the maximum degree of \(G\) respectively.
We generalize the techniques used in our algorithm for \(H\)-amalgamation recognition to the setting in which some of the graphs \(H,G_1,G_2,G\) are not given explicit at the input but are instead required to satisfy some topological property expressible in the counting monadic second order logic of graphs (CMSO logic). In this way, when restricting ourselves to graphs of constant treewidth and degree our approach generalizes certain algorithmic decomposition theorems from structural graph theory from the context of clique-sums to the context in which the interface graph \(H\) is given at the input.
For the entire collection see [Zbl 1398.68016].\(\forall\exists\mathbb {R}\)-completeness and area-universalityhttps://zbmath.org/1517.682902023-09-22T14:21:46.120933Z"Dobbins, Michael Gene"https://zbmath.org/authors/?q=ai:dobbins.michael-gene"Kleist, Linda"https://zbmath.org/authors/?q=ai:kleist.linda"Miltzow, Tillmann"https://zbmath.org/authors/?q=ai:miltzow.tillmann"Rzążewski, Paweł"https://zbmath.org/authors/?q=ai:rzazewski.pawelSummary: In the study of geometric problems, the complexity class \(\exists\mathbb{R}\) plays a crucial role since it exhibits a deep connection between purely geometric problems and real algebra. Sometimes \(\exists\mathbb {R}\) is referred to as the ``real analogue'' to the class NP. While NP is a class of computational problems that deals with existentially quantified Boolean variables, \(\exists\mathbb{R}\) deals with existentially quantified real variables.
In analogy to \(\varPi_2^p\) and \(\varSigma_2^p\) in the famous polynomial hierarchy, we study the complexity classes \(\forall\exists\mathbb{R}\) and \(\exists\forall\mathbb{R}\) with real variables. Our main interest is focused on the Area Universality problem, where we are given a plane graph \(G\), and ask if for each assignment of areas to the inner faces of \(G\) there is an area-realizing straight-line drawing of \(G\). We conjecture that the problem Area Universality is \(\forall\exists\mathbb{R}\)-complete and support this conjecture by a series of partial results, where we prove \(\exists\mathbb{R}\)- and \(\forall\exists\mathbb{R}\)-completeness of variants of Area Universality. To do so, we also introduce first tools to study \(\forall\exists\mathbb{R}\). Finally, we present geometric problems as candidates for \(\forall\exists\mathbb{R}\)-complete problems. These problems have connections to the concepts of imprecision, robustness, and extendability.
For the entire collection see [Zbl 1398.68016].Measuring what matters: a hybrid approach to dynamic programming with treewidthhttps://zbmath.org/1517.682912023-09-22T14:21:46.120933Z"Eiben, Eduard"https://zbmath.org/authors/?q=ai:eiben.eduard"Ganian, Robert"https://zbmath.org/authors/?q=ai:ganian.robert"Hamm, Thekla"https://zbmath.org/authors/?q=ai:hamm.thekla"Kwon, O-joung"https://zbmath.org/authors/?q=ai:kwon.ojoungSummary: We develop a framework for applying treewidth-based dynamic programming on graphs with ``hybrid structure'', i.e., with parts that may not have small treewidth but instead possess other structural properties. Informally, this is achieved by defining a refinement of treewidth which only considers parts of the graph that do not belong to a pre-specified tractable graph class. Our approach allows us to not only generalize existing fixed-parameter algorithms exploiting treewidth, but also fixed-parameter algorithms which use the size of a modulator as their parameter. As the flagship application of our framework, we obtain a parameter that combines treewidth and rank-width to obtain fixed-parameter algorithms for \textsc{Chromatic Number}, \textsc{Hamiltonian Cycle}, and \textsc{Max-Cut}.Measuring what matters: a hybrid approach to dynamic programming with treewidthhttps://zbmath.org/1517.682922023-09-22T14:21:46.120933Z"Eiben, Eduard"https://zbmath.org/authors/?q=ai:eiben.eduard"Ganian, Robert"https://zbmath.org/authors/?q=ai:ganian.robert"Hamm, Thekla"https://zbmath.org/authors/?q=ai:hamm.thekla"Kwon, O-Joung"https://zbmath.org/authors/?q=ai:kwon.ojoungSummary: We develop a framework for applying treewidth-based dynamic programming on graphs with ``hybrid structure'', i.e., with parts that may not have small treewidth but instead possess other structural properties. Informally, this is achieved by defining a refinement of treewidth which only considers parts of the graph that do not belong to a pre-specified tractable graph class. Our approach allows us to not only generalize existing fixed-parameter algorithms exploiting treewidth, but also fixed-parameter algorithms which use the size of a modulator as their parameter. As the flagship application of our framework, we obtain a parameter that combines treewidth and rank-width to obtain fixed-parameter algorithms for Chromatic Number, Hamiltonian Cycle, and Max-Cut.
For the entire collection see [Zbl 1423.68036].Atomic embeddability, clustered planarity, and thickenabilityhttps://zbmath.org/1517.682932023-09-22T14:21:46.120933Z"Fulek, Radoslav"https://zbmath.org/authors/?q=ai:fulek.radoslav"Tóth, Csaba D."https://zbmath.org/authors/?q=ai:toth.csaba-dVoronoi diagrams on planar graphs, and computing the diameter in deterministic \(\tilde{O}(n^{5/3})\) timehttps://zbmath.org/1517.682942023-09-22T14:21:46.120933Z"Gawrychowski, Paweł"https://zbmath.org/authors/?q=ai:gawrychowski.pawel"Kaplan, Haim"https://zbmath.org/authors/?q=ai:kaplan.haim"Mozes, Shay"https://zbmath.org/authors/?q=ai:mozes.shay"Sharir, Micha"https://zbmath.org/authors/?q=ai:sharir.micha"Weimann, Oren"https://zbmath.org/authors/?q=ai:weimann.orenThere is a deterministic \(\tilde{O}(n^{5/3})\)-time algorithm, where \(\tilde{O}\) hides polylogarithmic factors, for computing the diameter of a directed planar graph on \(n\) vertices and without negative-length cycles.
Reviewer: Haiko Müller (Leeds)Covering a graph with nontrivial vertex-disjoint paths: existence and optimizationhttps://zbmath.org/1517.682952023-09-22T14:21:46.120933Z"Gómez, Renzo"https://zbmath.org/authors/?q=ai:gomez.renzo"Wakabayashi, Yoshiko"https://zbmath.org/authors/?q=ai:wakabayashi.yoshikoSummary: Let \(G\) be a connected graph and \(\mathcal{P}\) be a set of pairwise vertex-disjoint paths in \(G\). We say that \(\mathcal{P}\) is a path cover if every vertex of \(G\) belongs to a path in \(\mathcal{P}\). In the Minimum Path Cover problem, one wishes to find a path cover of minimum cardinality. In this problem, known to be \({\textsc{NP}}\)-hard, the set \(\mathcal{P}\) may contain trivial (single-vertex) paths. We study the problem of finding a path cover composed only of nontrivial paths. First, we show that the Corresponding Existence problem can be reduced to a matching problem on a bipartite graph via the Edmonds-Gallai Decomposition. This reduction gives, in polynomial time, a certificate for both the yes-answer and the no-answer. When trivial paths are forbidden, for the feasible instances, one may consider either minimizing or maximizing the number of paths in the path cover. We show that the maximization problem has a close relation with the maximum matchings of a graph, and can be solved in polynomial time. For the minimization problem on feasible instances, we show that its computational complexity is equivalent to the Minimum Path Cover problem. We also show a linear-time algorithm on (edge-weighted) trees.
For the entire collection see [Zbl 1398.68016].Certificates and fast algorithms for biconnectivity in fully-dynamic graphshttps://zbmath.org/1517.682962023-09-22T14:21:46.120933Z"Henzinger, Monika R."https://zbmath.org/authors/?q=ai:rauch-henzinger.monika"La Poutré, Han"https://zbmath.org/authors/?q=ai:la-poutre.hanSummary: In this paper, we present sparse certificates for biconnectivity together with algorithms for updating these certificates. We thus obtain fully-dynamic algorithms for biconnectivity in graphs that run in \(O(\sqrt{n \log n} \log \lceil\frac{m}{n}\rceil)\) amortized time per operation, where \(m\) is the number of edges and \(n\) is the number of nodes in the graph. This improves upon the results in [\textit{M. Rauch}, STOC 1994, 686--695 (1994; Zbl 1344.68062)], in which algorithms were presented running in \(O(\sqrt{m})\) amortized time, and solves the open problem to find certificates to speed up biconnectivity, as stated in [\textit{D. Eppstein} et al., FOCS 1992, 60--69 (1992; Zbl 0977.68560); J. ACM 44, No. 5, 669--696 (1997; Zbl 0891.68072)].
For the entire collection see [Zbl 0854.00022].Connectivity and minimum cut approximation in the broadcast congested cliquehttps://zbmath.org/1517.682972023-09-22T14:21:46.120933Z"Jurdziński, Tomasz"https://zbmath.org/authors/?q=ai:jurdzinski.tomasz"Nowicki, Krzysztof"https://zbmath.org/authors/?q=ai:nowicki.krzysztofSummary: In this paper we present two graph algorithms in the broadcast congested clique model. In this model, there are \(n\) players, which communicate in synchronous rounds. Each player represents a single node of the input graph; initially each player knows the set of edges incident to his node. In each round of communication each node can broadcast a single \(b\)-bit message to all other nodes; usually \(b\in{\mathcal{O}}(\log n)\). The goal is to compute some function of the input graph.
The first result we present is the first sub-logarithmic deterministic algorithm finding a maximal spanning forest of an \(n\) node graph in the broadcast congested clique, which requires only \({\mathcal{O}}(\log n/\log\log n)\) rounds. The second result is a randomized \(1+\epsilon\) approximation algorithm finding the minimum cut of an \(n\) node graph, which requires only \({\mathcal{O}}(\log n)\) maximal spanning forest computations. In the broadcast congested clique this approach, combined with the new maximal spanning forest algorithm, yields an \({\mathcal{O}}(\log^2 n/\log\log n)\) round algorithm. Additionally, it may be applied to different models, i.e. in the multi-pass semi-streaming model it allows to reduce required memory by \(\varTheta(\log n)\) factor, with only \({\mathcal{O}}(\log^* n)\) passes over the data stream.
For the entire collection see [Zbl 1400.68027].Convexity-increasing morphs of planar graphshttps://zbmath.org/1517.682982023-09-22T14:21:46.120933Z"Kleist, Linda"https://zbmath.org/authors/?q=ai:kleist.linda"Klemz, Boris"https://zbmath.org/authors/?q=ai:klemz.boris"Lubiw, Anna"https://zbmath.org/authors/?q=ai:lubiw.anna"Schlipf, Lena"https://zbmath.org/authors/?q=ai:schlipf.lena"Staals, Frank"https://zbmath.org/authors/?q=ai:staals.frank"Strash, Darren"https://zbmath.org/authors/?q=ai:strash.darrenSummary: We study the problem of convexifying drawings of planar graphs. Given any planar straight-line drawing of a 3-connected graph \(G\), we show how to morph the drawing to one with convex faces while maintaining planarity at all times. Furthermore, the morph is convexity increasing, meaning that angles of inner faces never change from convex to reflex. We give a polynomial time algorithm that constructs such a morph as a composition of a linear number of steps where each step either moves vertices along horizontal lines or moves vertices along vertical lines.
For the entire collection see [Zbl 1398.68016].The component (edge) connectivity of round matching composition networkshttps://zbmath.org/1517.683002023-09-22T14:21:46.120933Z"Liu, Xiaoqing"https://zbmath.org/authors/?q=ai:liu.xiaoqing"Zhou, Shuming"https://zbmath.org/authors/?q=ai:zhou.shuming"Zhang, Hong"https://zbmath.org/authors/?q=ai:zhang.hong.13"Niu, Baohua"https://zbmath.org/authors/?q=ai:niu.baohuaSummary: The vertex (edge) connectivity has been regularly used to measure the fault tolerance and reliability of interconnection networks, while it has defects in the assumption that all neighbors of one node will fail concurrently. To overcome this deficiency, some new generalizations of traditional connectivity have been suggested to quantize the size or the number of the connected components of the survival graph. The \(g\)-component (edge) connectivity, one generalization of vertex (edge) connectivity, has been proposed to characterize the vulnerability of multiprocessor systems based on the number of components of the survival graph. In this paper, we determine the \(g\)-component (edge) connectivity of a family of networks, called the round matching composition networks RMCNs, which are a class of networks composed of \(t\) \((t\geq 4)\) clusters with the same order, linked by \(r\) perfect matchings. By exploring the combinatorial properties and fault-tolerance of RMCNs, we establish the \(g\)-component (edge) connectivity \(c\kappa_{g+1}(G)=gk- g^2+3g\) for \(2\leq g\leq k+3\) and \(k\geq 2\), \(c\lambda_3(G)=2k+3\) and \(c\lambda_4(G)=3k+4\) for \(k\geq 4\).Construction and local routing for angle-monotone graphshttps://zbmath.org/1517.683012023-09-22T14:21:46.120933Z"Lubiw, Anna"https://zbmath.org/authors/?q=ai:lubiw.anna"Mondal, Debajyoti"https://zbmath.org/authors/?q=ai:mondal.debajyotiSummary: A geometric graph in the plane is angle-monotone of width \(\gamma\) if every pair of vertices is connected by an angle-monotone path of width \(\gamma\), a path such that the angles of any two edges in the path differ by at most \(\gamma\). Angle-monotone graphs have good spanning properties.
We prove that every point set in the plane admits an angle-monotone graph of width \(90^\circ\), hence with spanning ratio \(\sqrt{2}\), and a subquadratic number of edges. This answers an open question posed by \textit{H. R. Dehkordi} et al. [J. Graph Algorithms Appl. 19, No. 2, 761--778 (2015; Zbl 1328.05054)].
We show how to construct, for any point set of size \(n\) and any angle \(\alpha\), \(0<\alpha<45^\circ\), an angle-monotone graph of width \((90^\circ+\alpha)\) with \(O(\frac{n}{\alpha})\) edges. Furthermore, we give a local routing algorithm to find angle-monotone paths of width \((90^\circ+\alpha)\) in these graphs. The routing ratio, which is the ratio of path length to Euclidean distance, is at most \(1/\cos(45^\circ+\frac{\alpha}{2})\), i.e., ranging from \(\sqrt{2}\approx 1.414\) to 2.613. For the special case \(\alpha=30^\circ\), we obtain the \(\varTheta_6\)-graph and our routing algorithm achieves the known routing ratio 2 while finding angle-monotone paths of width \(120^\circ\).
For the entire collection see [Zbl 1398.68016].Approximability of open \(k\)-monopoly problemshttps://zbmath.org/1517.683022023-09-22T14:21:46.120933Z"Mishra, Sounaka"https://zbmath.org/authors/?q=ai:mishra.sounaka"Krishna, B. Arjuna"https://zbmath.org/authors/?q=ai:krishna.b-arjuna"Rajakrishnan, Shijin"https://zbmath.org/authors/?q=ai:rajakrishnan.shijinSummary: We consider approximability of two optimization problems called Minimum Open \(k\)-Monopoly (Min-Open-\(k\)-Monopoly) and Minimum Partial Open \(k\)-Monopoly (Min-P-Open-\(k\)-Monopoly), where \(k\) is a fixed positive integer. The objective, in Min-Open-\(k\)-Monopoly, is to find a minimum cardinality vertex set \(S\subseteq V\) in a given graph \(G=(V,E)\) such that \(|N(v)\cap S|\geq\frac{1}{2}|N(v)|+k\), for every vertex \(v \in V\). On the other hand, given a graph \(G=(V,E)\), in Min-P-Open-\(k\)-Monopoly it is required to find a minimum cardinality vertex set \(S\subseteq V\) such that \(|N(v)\cap S|\geq\frac{1}{2} |N(v)|+k\), for every \(v\in V\setminus S\). We prove that Min-Open-\(k\)-Monopoly and Min-P-Open-\(k\)-Monopoly are approximable within a factor of \(O(\log n)\). Then, we show that these two problems cannot be approximated within a factor of \((\frac{1}{3}- \epsilon)\ln n\) and \((\frac{1}{4}-\epsilon)\ln n\), respectively, for any \(\varepsilon>0\), unless \(\mathsf{NP}\subseteq\mathsf{Dtime}(n^{O(\log\log n)})\). For 4-regular graphs, we prove that these two problems are \textsf{APX}-complete. Min-Open-1-Monopoly can be approximated within a factor of \(\frac{26}{21}\approx 1.2381\) where as Min-P-Open-1-Monopoly can be approximated within a factor of 1.65153. For \(k\geq 2\), we also present approximation algorithms for these two problems for \((2k+2)\)-regular graphs.On the maximum edge-pair embedding bipartite matchinghttps://zbmath.org/1517.683052023-09-22T14:21:46.120933Z"Nguyen, Cam Ly"https://zbmath.org/authors/?q=ai:nguyen.cam-ly"Suppakitpaisarn, Vorapong"https://zbmath.org/authors/?q=ai:suppakitpaisarn.vorapong"Surarerks, Athasit"https://zbmath.org/authors/?q=ai:surarerks.athasit"Vajanopath, Phanu"https://zbmath.org/authors/?q=ai:vajanopath.phanuSummary: Given a set of edge pairs in a bipartite graph, we want to find a bipartite matching that includes a maximum number of those edge pairs. While the problem has many applications to wireless localization, to the best of our knowledge, there is no theoretical work for the problem. In this work, unless \(\mathrm{P = NP}\), we show that there is no constant approximation for the problem. Suppose that \(k\) denotes the maximum number of input edge pairs such that a particular node can be in. Inspired by experimental results, we consider the case that \(k\) is small. While there is a simple polynomial-time algorithm for the problem when \(k\) is one, we show that the problem is NP-hard when \(k\) is greater than one. We also devise an efficient \(O(k)\)-approximation algorithm for the problem.
For the entire collection see [Zbl 1435.68040].On the maximum edge-pair embedding bipartite matchinghttps://zbmath.org/1517.683062023-09-22T14:21:46.120933Z"Nguyen, Cam Ly"https://zbmath.org/authors/?q=ai:nguyen.cam-ly"Suppakitpaisarn, Vorapong"https://zbmath.org/authors/?q=ai:suppakitpaisarn.vorapong"Surarerks, Athasit"https://zbmath.org/authors/?q=ai:surarerks.athasit"Vajanopath, Phanu"https://zbmath.org/authors/?q=ai:vajanopath.phanuSummary: Given a set of edge pairs in a complete bipartite graph, we want to find a bipartite matching that includes the maximum number of those edge pairs. While the problem has many applications to wireless localization and computer vision, to the best of our knowledge, there is no theoretical work for the problem. In this work, unless \(\mathrm{P = NP}\), we show that there is no constant approximation for the problem. Then, we consider two special cases of the problem. Suppose that \(k\) denotes the maximum number of input edge pairs such that a particular node can be in. Inspired by experimental results, the first case is for when \(k\) is not large. While there is a simple polynomial-time algorithm for the problem when \(k\) is one, we show that the problem is NP-hard when \(k\) is greater than one. We also devise an efficient \(O(k)\)-approximation algorithm for the problem. For the second case, every pair of nodes in the same partition of the input bipartite graph are labeled with one of \(\chi\) colors. We want to match, between the two partitions, a pair of nodes to a pair of nodes with the same color. Denote \(n\) as the number of nodes, we give an \(O(\sqrt{ \chi n})\)-approximation algorithm for this case.Explorable families of graphshttps://zbmath.org/1517.683072023-09-22T14:21:46.120933Z"Pelc, Andrzej"https://zbmath.org/authors/?q=ai:pelc.andrzejSummary: Graph exploration is one of the fundamental tasks performed by a mobile agent in a graph. An \(n\)-node graph has unlabeled nodes, and all ports at any node of degree \(d\) are arbitrarily numbered \(0,\dots,d-1\). A mobile agent, initially situated at some starting node \(v\), has to visit all nodes of the graph and stop. In the absence of any initial knowledge of the graph the task of deterministic exploration is often impossible. On the other hand, for some families of graphs it is possible to design deterministic exploration algorithms working for any graph of the family. We call such families of graphs explorable. Examples of explorable families are all finite families of graphs, as well as the family of all trees.
In this paper we study the problem of which families of graphs are explorable. We characterize all such families, and then ask the question whether there exists a universal deterministic algorithm that, given an explorable family of graphs, explores any graph of this family, without knowing which graph of the family is being explored. The answer to this question turns out to depend on how the explorable family is given to the hypothetical universal algorithm. If the algorithm can get the answer to any yes/no question about the family, then such a universal algorithm can be constructed. If, on the other hand, the algorithm can be only given an algorithmic description of the input explorable family, then such a universal deterministic algorithm does not exist.
For the entire collection see [Zbl 1400.68027].Faster parameterized algorithm for cluster vertex deletionhttps://zbmath.org/1517.683082023-09-22T14:21:46.120933Z"Tsur, Dekel"https://zbmath.org/authors/?q=ai:tsur.dekelSummary: In the \textsc{Cluster Vertex Deletion} problem the input is a graph \(G\) and an integer \(k\). The goal is to decide whether there is a set of vertices \(S\) of size at most \(k\) such that the deletion of the vertices of \(S\) from \(G\) results in a graph in which every connected component is a clique. We give an algorithm for \textsc{Cluster Vertex Deletion} whose running time is \(O^\ast (1.811^k)\).Fault-tolerant strong Menger (edge) connectivity of DCC linear congruential graphshttps://zbmath.org/1517.683092023-09-22T14:21:46.120933Z"Yu, Zhengqin"https://zbmath.org/authors/?q=ai:yu.zhengqin"Zhou, Shuming"https://zbmath.org/authors/?q=ai:zhou.shuming"Zhang, Hong"https://zbmath.org/authors/?q=ai:zhang.hong.13Summary: With the rapid development and advances of very large scale integration technology and wafer-scale integration technology, multiprocessor systems, taking interconnection networks as underlying topologies, have been widely designed and used in big data era. The topology of an interconnection network is usually represented as a graph. If any two distinct vertices \(x,y\) in a connected graph \(G\) are connected by \(\min\{\deg_G(x),\deg_G(y)\}\) vertex (edge)-disjoint paths, then \(G\) is called strongly Menger (edge) connected.
In [IEEE Trans. Comput. 45, No. 2, 156--164 (1996; Zbl 1068.68559)], \textit{J. Opatrny} et al. introduced the DCC (Disjoint Consecutive Cycle) linear congruential graph, which consists of \(n\) nodes and is generated by a set of linear functions \(F\) with special properties. In this work, we investigate the strong Menger connectivity of the DCC linear congruential graph \(\mathcal{G}= G_{2t}(F,2^p)\) with faulty vertices or edges, where \(2\leq t\leq p-1\), \(p\geq 5\), \(F=\{f_i(x)\mid f_i(x)=(2^{i-1}b+1)x+2^{i-1}c,1\leq i\leq t\}\), \(\gcd(c,2^p)=1\) and \(b\) is a multiple of \(4\). In detail, we show that \(\mathcal{G}-S\) is strongly Menger connected if \(|S|\leq 2t-4\) for any \(S\subseteq V(\mathcal{G})\). Moreover, we determine that \(\mathcal{G}-S\) is strongly Menger edge connected if \(|S|\leq 2t-2\) for any \(S\subseteq E(\mathcal{G})\). Furthermore, we prove that, under the restricted condition \(\delta(\mathcal{G}-S)\geq 2\), \(\mathcal{G}-S\) is strongly Menger edge connected if \(|S|\leq 4t-6\) and \(t\geq 3\) for any \(S\subseteq E(\mathcal{G})\). In addition, we present some empirical examples to show that the bounds are all optimal in the sense of the maximum number of tolerable edge faults.Structure and substructure connectivity of divide-and-swap cubehttps://zbmath.org/1517.683102023-09-22T14:21:46.120933Z"Zhou, Qianru"https://zbmath.org/authors/?q=ai:zhou.qianru"Zhou, Shuming"https://zbmath.org/authors/?q=ai:zhou.shuming"Liu, Jiafei"https://zbmath.org/authors/?q=ai:liu.jiafei"Liu, Xiaoqing"https://zbmath.org/authors/?q=ai:liu.xiaoqingSummary: High fault tolerance and reliability of multiprocessor systems, modeled by interconnection network, are of great significance to assess the flexibility and effectiveness of the systems. Connectivity is an important metric to evaluate the fault tolerance and reliability of interconnection networks. As classical connectivity is not suitable for such large scale systems, a novel and generalized connectivity, structure connectivity and substructure connectivity, has been proposed to measure the robustness of networks and has witnessed rich achievements. The divide-and-swap cube \(\mathit{DSC}_n\) is an interesting variant of hypercube that has nice hierarchical properties. In this paper, we mainly investigate \(\mathcal{H} \)-structure-connectivity, denoted by \(\kappa(\mathit{DSC}_n; \mathcal{H})\), and \(\mathcal{H} \)-substructure-connectivity, denoted by \(\kappa^s(\mathit{DSC}_n; \mathcal{H})\), for \(\mathcal{H} \in \{ K_1, K_{1 , 1}, K_{1 , m}(2 \leq m \leq d + 1), C_4 \} \), respectively. In detail, we show that \(\kappa(\mathit{DSC}_n; K_1) = \kappa^s(\mathit{DSC}_n; K_1) = d + 1\) for \(n \geq 2\), \(\kappa(\mathit{DSC}_n; K_{1 , 1}) = \kappa^s(\mathit{DSC}_n; K_{1 , 1}) = d + 1\) for \(n \geq 8\), \(\kappa(\mathit{DSC}_n; K_{1 , m}) = \kappa^s(\mathit{DSC}_n; K_{1 , m}) = \lfloor \frac{ d}{ 2} \rfloor + 1\) with \(2 \leq m \leq d + 1\) for \(n \geq 4\), \(\kappa(\mathit{DSC}_n; C_4) = 3 + 2(d - 2)\) for \(4 \leq n \leq 8\), \(\lfloor \frac{ d}{ 2} \rfloor + 1 \leq \kappa(\mathit{DSC}_n; C_4) \leq d + 1\) for \(n \geq 16\) and \(\kappa^s(\mathit{DSC}_n; C_4) = \lfloor \frac{ d}{ 2} \rfloor + 1\) for \(n \geq 4\). Finally, we compare and analyze the ratios of structure (resp. substructure) connectivity to vertex degree of divide-and-swap cube with that of several well-known variants of hypercube.Statistics on bargraphs of Catalan wordshttps://zbmath.org/1517.683112023-09-22T14:21:46.120933Z"Callan, David"https://zbmath.org/authors/?q=ai:callan.david"Mansour, Toufik"https://zbmath.org/authors/?q=ai:mansour.toufik"Ramírez, José L."https://zbmath.org/authors/?q=ai:ramirez.jose-luisSummary: Catalan words are a particular case of growth-restricted words. Here we give the bivariate generating function for bargraphs associated to Catalan words according to the semiperimeter and area statistics. Exact formulas to count Catalan bargraphs according to the two statistics are also found. We show a connection to the Narayana numbers. Finally, we give similar results for the exterior corner statistic.Corrigendum to: ``Parallel algorithm for solving the graph isomorphism problem''https://zbmath.org/1517.684082023-09-22T14:21:46.120933Z"Vasilchikov, V. V."https://zbmath.org/authors/?q=ai:vasilchikov.vladimir-vasilevichFrom the text: In the article by the author [ibid. 27, No. 1, 86--94 (2020; Zbl 1513.68079)] there was a misprint in the layout. In the Table 1, in the last column of the row ``Degree of graph'' the value should be 3000 (instead of 300). The corrected ``Table 1'' is shown below. The editors apologise for the inconvenience.Deterministic distributed dominating set approximation in the CONGEST modelhttps://zbmath.org/1517.684112023-09-22T14:21:46.120933Z"Deurer, Janosch"https://zbmath.org/authors/?q=ai:deurer.janosch"Kuhn, Fabian"https://zbmath.org/authors/?q=ai:kuhn.fabian"Maus, Yannic"https://zbmath.org/authors/?q=ai:maus.yannicDiscontinuous Galerkin method with Voronoi partitioning for quantum simulation of chemistryhttps://zbmath.org/1517.810382023-09-22T14:21:46.120933Z"Faulstich, Fabian M."https://zbmath.org/authors/?q=ai:faulstich.fabian-m"Wu, Xiaojie"https://zbmath.org/authors/?q=ai:wu.xiaojie"Lin, Lin"https://zbmath.org/authors/?q=ai:lin.linSummary: To circumvent a potentially dense two-body interaction tensor and obtain lower asymptotic costs for quantum simulations of chemistry, the discontinuous Galerkin (DG) basis set with a rectangular partitioning strategy was recently introduced [\textit{J. R. McClean} et al, New J. Phys. 22, No. 9, Article ID 093015, 25 p. (2020; \url{doi:10.1088/1367-2630/ab9d9f})]. We propose and numerically scrutinize a more general DG basis set construction based on a Voronoi decomposition with respect to the nuclear coordinates. This allows the construction of DG basis sets for arbitrary molecular and crystalline configurations. We here employ the planewave dual basis set as primitive basis set in the supercell model; as a set of grid-based nascent delta functions, the planewave dual functions provide sufficient flexibility for the Voronoi partitioning. The presented implementation of this \textit{DG-Voronoi} approach is in Python and solely based on PySCF. We numerically investigate the performance, at the mean-field and correlated level of theory for quasi-1D, quasi-2D and fully 3D systems, and exemplify the application to crystalline systems.Band structure of the one-dimensional spin-orbit-coupled Su-Schrieffer-Heeger lattice with \(\mathcal{PT}\)-symmetric onsite imaginary potentialshttps://zbmath.org/1517.810592023-09-22T14:21:46.120933Z"Li, Jia-Rui"https://zbmath.org/authors/?q=ai:li.jiarui"Wang, Zi-An"https://zbmath.org/authors/?q=ai:wang.zian"Zhang, Lian-Lian"https://zbmath.org/authors/?q=ai:zhang.lian-lianSummary: Energy band structures of one-dimensional non-Hermitian spin-orbit-coupled Su-Schrieffer-Heeger (SSH) model are theoretically investigated, by introducing imaginary potentials with gain and loss effects. It is found that the imaginary potentials can promote the occurrence of \(\mathcal{PT}\)-symmetry breaking. In the topologically-nontrivial regions, the different imaginary parts of the edge-state energies leads to the different localization degrees of such states. Moreover, gapless phase regions arise between the topologically-nontrivial and -trivial regions, in which edge states are allowed to survive. Therefore, these results are helpful to understand the band-structural and topological properties of \(\mathcal{PT}\)-symmetric non-Hermitian systems.Semiclassical calculation of time delay statistics in chaotic quantum scatteringhttps://zbmath.org/1517.810612023-09-22T14:21:46.120933Z"Novaes, Marcel"https://zbmath.org/authors/?q=ai:novaes.marcelSummary: We present a semiclassical calculation, based on classical action correlations implemented by means of a matrix integral, of all moments of the Wigner-Smith time delay matrix, \(Q\), in the context of quantum scattering through systems with chaotic dynamics. Our results are valid for broken time reversal symmetry and depend only on the classical dwell time and the number of open channels, \(M\), which is arbitrary. Agreement with corresponding random matrix theory reduces to an identity involving some combinatorial concepts, which can be proved in special cases.Spectrum of the hypereclectic spin chain and Pólya countinghttps://zbmath.org/1517.810802023-09-22T14:21:46.120933Z"Ahn, Changrim"https://zbmath.org/authors/?q=ai:ahn.changrim"Staudacher, Matthias"https://zbmath.org/authors/?q=ai:staudacher.matthiasSummary: In earlier work we proposed a generating function that encodes the Jordan block spectrum of the integrable Hypereclectic spin chain, related to the one-loop dilatation operator of the dynamical fishnet quantum field theory. We significantly improve the expressions for these generating functions, rendering them much more explicit and elegant. In particular, we treat the case of the full spin chain without imposing any cyclicity constraints on the states, as well as the case of cyclic states. The latter involves the Pólya enumeration theorem in conjunction with \(q\)-binomial coefficients.Weak multiplex percolationhttps://zbmath.org/1517.820032023-09-22T14:21:46.120933Z"Baxter, Gareth J."https://zbmath.org/authors/?q=ai:baxter.gareth-j"da Costa, Rui A."https://zbmath.org/authors/?q=ai:da-costa.rui-a"Dorogovtsev, Sergey N."https://zbmath.org/authors/?q=ai:dorogovtsev.sergey-n"Mendes, José F. F."https://zbmath.org/authors/?q=ai:mendes.jose-f-fPublisher's description: In many systems consisting of interacting subsystems, the complex interactions between elements can be represented using multilayer networks. However percolation, key to understanding connectivity and robustness, is not trivially generalised to multiple layers. This Element describes a generalisation of percolation to multilayer networks: weak multiplex percolation. A node belongs to a connected component if at least one of its neighbours in each layer is in this component. The authors fully describe the critical phenomena of this process. In two layers with finite second moments of the degree distributions the authors observe an unusual continuous transition with quadratic growth above the threshold. When the second moments diverge, the singularity is determined by the asymptotics of the degree distributions, creating a rich set of critical behaviours. In three or more layers the authors find a discontinuous hybrid transition which persists even in highly heterogeneous degree distributions, becoming continuous only when the powerlaw exponent reaches \(1+1/(M-1)\) for \(M\) layers.Free boundary dimers: random walk representation and scaling limithttps://zbmath.org/1517.820102023-09-22T14:21:46.120933Z"Berestycki, Nathanaël"https://zbmath.org/authors/?q=ai:berestycki.nathanael"Lis, Marcin"https://zbmath.org/authors/?q=ai:lis.marcin"Qian, Wei"https://zbmath.org/authors/?q=ai:qian.weiSummary: We study the dimer model on subgraphs of the square lattice in which vertices on a prescribed part of the boundary (the free boundary) are possibly unmatched. Each such unmatched vertex is called a monomer and contributes a fixed multiplicative weight \(z>0\) to the total weight of the configuration. A bijection described by \textit{A. Giuliani} et al. [J. Stat. Phys. 163, No. 2, 211--238 (2016; Zbl 1342.82161)] relates this model to a standard dimer model but on a non-bipartite graph. The Kasteleyn matrix of this dimer model describes a walk with transition weights that are negative along the free boundary. Yet under certain assumptions, which are in particular satisfied in the infinite volume limit in the upper half-plane, we prove an effective, true random walk representation for the inverse Kasteleyn matrix. In this case we further show that, independently of the value of \(z>0\), the scaling limit of the centered height function is the Gaussian free field with Neumann (or free) boundary conditions. It is the first example of a discrete model where such boundary conditions arise in the continuum scaling limit.Universality of spin correlations in the Ising model on isoradial graphshttps://zbmath.org/1517.820122023-09-22T14:21:46.120933Z"Chelkak, Dmitry"https://zbmath.org/authors/?q=ai:chelkak.dmitry"Izyurov, Konstantin"https://zbmath.org/authors/?q=ai:izyurov.konstantin"Mahfouf, Rémy"https://zbmath.org/authors/?q=ai:mahfouf.remySummary: We prove universality of spin correlations in the scaling limit of the planar Ising model on isoradial graphs with uniformly bounded angles and Z-invariant weights. Specifically, we show that in the massive scaling limit, that is, as the mesh size tends to zero at the same rate as the Baxter elliptic parameter tends to 1, the two-point spin correlations in the full plane converge to a universal rotationally invariant limit.
These results, together with techniques developed to obtain them, are sufficient to extend to isoradial graphs, the convergence results for multipoint spin correlations in bounded planar domains which were previously known only on the square grid. We also give a simple proof of the fact that the infinite-volume magnetization in a subcritical Z-invariant Ising model is independent of the site and of the lattice.
As compared to techniques already existing in the literature, we streamline the analysis of discrete (massive) holomorphic spinors near their ramification points which also provides a solid ground for further generalizations.Fluctuations in mean-field Ising modelshttps://zbmath.org/1517.820132023-09-22T14:21:46.120933Z"Deb, Nabarun"https://zbmath.org/authors/?q=ai:deb.nabarun"Mukherjee, Sumit"https://zbmath.org/authors/?q=ai:mukherjee.sumitSummary: In this paper, we study the fluctuations of the average magnetization in an Ising model on an approximately \(d_N\) regular graph \(G_N\) on \(N\) vertices. In particular, if \(G_N\) satisfies a ``spectral gap'' condition, we show that whenever \(d_N \gg \sqrt{N}\), the fluctuations are universal and the same as that of the Curie-Weiss model in the entire ferromagnetic parameter regime. We give a counterexample to demonstrate that the condition \(d_N \gg \sqrt{N}\) is tight, in the sense that the limiting distribution changes if \(d_N \sim \sqrt{N}\) except in the high temperature regime. By refining our argument, we extend universality in the high temperature regime up to \(d_N \gg N^{1/3}\). Our results include universal fluctuations of the average magnetization in Ising models on regular graphs, Erdős-Rényi graphs (directed and undirected), stochastic block models, and sparse regular graphons. In fact, our results apply to general matrices with nonnegative entries, including Ising models on a Wigner matrix, and the block spin Ising model. As a by-product of our proof technique, we obtain Berry-Esseen bounds for these fluctuations, exponential concentration for the average of spins, tight error bounds for the mean-field approximation of the partition function, and tail bounds for various statistics of interest.Uniqueness of the Gibbs measure for the 4-state anti-ferromagnetic Potts model on the regular treehttps://zbmath.org/1517.820142023-09-22T14:21:46.120933Z"De Boer, David"https://zbmath.org/authors/?q=ai:de-boer.david"Buys, Pjotr"https://zbmath.org/authors/?q=ai:buys.pjotr"Regts, Guus"https://zbmath.org/authors/?q=ai:regts.guusSummary: We show that the \(4\)-state anti-ferromagnetic Potts model with interaction parameter \(w\in (0,1)\) on the infinite \((d+1)\)-regular tree has a unique Gibbs measure if \(w\geq 1-\dfrac{4}{d+1}\) for all \(d\geq 4\). This is tight since it is known that there are multiple Gibbs measures when \(0\leq w< 1-\dfrac{4}{d+1}\) and \(d\geq 4\). We moreover give a new proof of the uniqueness of the Gibbs measure for the \(3\)-state Potts model on the \((d+1)\)-regular tree for \(w\geq 1-\dfrac{3}{d+1}\) when \(d\geq 3\) and for \(w\in (0,1)\) when \(d=2\) .Gibbs measures for HC-model with a cuountable set of spin values on a Cayley treehttps://zbmath.org/1517.820202023-09-22T14:21:46.120933Z"Khakimov, R. M."https://zbmath.org/authors/?q=ai:khakimov.rustam-m"Makhammadaliev, M. T."https://zbmath.org/authors/?q=ai:makhammadaliev.mukhtorjon-tursunmukhammad-ogli"Rozikov, U. A."https://zbmath.org/authors/?q=ai:rozikov.utkir-aSummary: In this paper, we study the HC-model with a countable set \(\mathbb{Z}\) of spin values on a Cayley tree of order \(k \geq 2\). This model is defined by a countable set of parameters (that is, the activity function \(\lambda_i > 0\), \(i\in\mathbb{Z}\)). A functional equation is obtained that provides the consistency condition for finite-dimensional Gibbs distributions. Analyzing this equation, the following results are obtained:
\begin{itemize}
\item[--] Let \(\Lambda = \sum_i\lambda_i\). For \(\Lambda = +\infty\) there is no translation-invariant Gibbs measure (TIGM) and no two-periodic Gibbs measure (TPGM);
\item[--] For \(\Lambda < +\infty\), the uniqueness of TIGM is proved;
\item[--] Let \(\Lambda_{\mathrm{cr}}(k) = \frac{k^k}{(k-1)^{k+1}}\). If \(0 < \Lambda \leq \Lambda_{\mathrm{cr}}\), then there is exactly one TPGM that is TIGM;
\item[--] For \(\Lambda > \Lambda_{\mathrm{cr}}\), there are exactly three TPGMs, one of which is TIGM.
\end{itemize}Convergence of the Fefferman-Graham expansion and complex black hole anatomyhttps://zbmath.org/1517.830512023-09-22T14:21:46.120933Z"Serantes, Alexandre"https://zbmath.org/authors/?q=ai:serantes.alexandre"Withers, Benjamin"https://zbmath.org/authors/?q=ai:withers.benjaminSummary: Given a set of sources and one-point function data for a Lorentzian holographic QFT, does the Fefferman-Graham expansion converge? If it does, what sets the radius of convergence, and how much of the interior of the spacetime can be reconstructed using this expansion? As a step towards answering these questions we consider real analytic conformal field theory data, where in the absence of logarithms, the radius is set by singularities of the complex metric reached by analytically continuing the Fefferman-Graham radial coordinate. With the conformal boundary at the origin of the complex radial plane, real Lorentzian submanifolds appear as piecewise paths built from radial rays and arcs of circles centred on the origin. This allows singularities of Fefferman-Graham metric functions to be identified with gauge-invariant singularities of maximally extended black hole spacetimes, thereby clarifying the physical cause of the limited radius of convergence in such cases. We find black holes with spacelike singularities can give a radius of convergence equal to the horizon radius, however for black holes with timelike singularities the radius is smaller. We prove that a finite radius of convergence does not necessarily follow from the existence of an event horizon, a spacetime singularity, nor from caustics of the Fefferman-Graham gauge, by providing explicit examples of spacetimes with an infinite radius of convergence which contain such features.Combinatorial sequences for disaster scenario generationhttps://zbmath.org/1517.900112023-09-22T14:21:46.120933Z"Garn, Bernhard"https://zbmath.org/authors/?q=ai:garn.bernhard"Kieseberg, Klaus"https://zbmath.org/authors/?q=ai:kieseberg.klaus"Schreiber, Dominik"https://zbmath.org/authors/?q=ai:schreiber.dominik"Simos, Dimitris E."https://zbmath.org/authors/?q=ai:simos.dimitris-eSummary: Training exercises are an important tool in crisis management, as they can assist in a multitude of tasks, such as planning pre-crisis resource requirements and allocation, response planning and help train emergency personnel for actual crises. To be effective, the exercises have to utilize well constructed scenarios and be able to replicate certain characteristics of a crisis situation. In this paper, we propose a conceptual mathematical modeling approach for the automated generation of scenarios for disaster exercises via certain combinatorial sequence structures. The derived scenarios within an exercise collectively fulfill different notions of combinatorial sequence coverage, thereby providing the means to test existing response strategies for deficiencies as well as to train emergency personnel for their ability to handle different arrangements of events. This guaranteed diversity by construction can be used as a basis to obtain quantitative assurance statements when these scenarios have been successfully mastered by participants in exercises. We illustrate our proposed approach utilizing two different combinatorial structures for two example disasters.Leader-follower coherence in noisy ring-trees networkshttps://zbmath.org/1517.900232023-09-22T14:21:46.120933Z"Sun, Weigang"https://zbmath.org/authors/?q=ai:sun.weigang"Hong, Meidu"https://zbmath.org/authors/?q=ai:hong.meidu"Liu, Suyu"https://zbmath.org/authors/?q=ai:liu.suyu"Fan, Kai"https://zbmath.org/authors/?q=ai:fan.kaiSummary: This paper investigates leader-follower network coherence in a noisy ring-trees network model with preassigned leaders at the initial state. Different from existing works on designing consensus algorithms in the multi-agent systems, the leader-follower coherence characterized by the eigenvalues of a principal submatrix obtained from the Laplacian matrix is a measure of deviation from the state of the leaders in an \(H_2\) norm. The recursive properties of ring-trees networks allow analytical calculations of this network coherence. Based on the relationship of the eigenvalues of the submatrix in two successive steps, an analytical expression for the leader-follower coherence is determined depending on the number of leaders and network parameters. This network model shows better consensus with the increasing number of leaders in the ring network and the ring-trees topology has a profound impact on the coherence.Nonexistence of uniformly most reliable two-terminal graphshttps://zbmath.org/1517.900332023-09-22T14:21:46.120933Z"Xie, Sun"https://zbmath.org/authors/?q=ai:xie.sun"Zhao, Haixing"https://zbmath.org/authors/?q=ai:zhao.haixing"Yin, Jun"https://zbmath.org/authors/?q=ai:yin.jun|yin.jun.1Summary: A two-terminal graph \(G = (V, E)\) is a simple and undirected graph with two specified target vertices \(s\) and \(t\) in \(V\). In \(G\), if each edge survives independently with a fixed probability \(p\), the two-terminal reliability is the probability that two target vertices are connected. A two-terminal graph is uniformly most reliable if its reliability is not less than the reliability of any other graph with same number of vertices and edges for all \(p\). Betrand et al. proved that there is no uniformly most reliable two-terminal graph if either \(n \geq 11\) and \(20 \leq m \leq 3 n - 9\) or \(n \geq 8\) and \(\binom{n}{2} - \lfloor(n - 2) / 2 \rfloor \leq m \leq \binom{n}{2} - 2\). In this paper, we further prove that there is no uniformly most reliable two-terminal graph if \(n \geq 6\) and \(3 n - 6 < m \leq \binom{n}{2} - 2\) in a different way.Applying iterated mapping to the no-three-in-a-line problemhttps://zbmath.org/1517.901102023-09-22T14:21:46.120933Z"Brower, Cole"https://zbmath.org/authors/?q=ai:brower.cole-rubin"Ponomarenko, Vadim"https://zbmath.org/authors/?q=ai:ponomarenko.vadimSummary: Iterated mapping has seen a lot of success lately in many problems such as bit retrieval, diffraction signal reconstruction, and graph coloring. We add another application of iterated mapping, namely finding solutions to the no-three-in-a-line problem. Given an \(n \times n\) grid, we utilize iterated mapping to find \(2n\) points such that any straight line (of any slope) drawn will not intersect three of the selected points.Inverse 1-median problem on trees under mixed rectilinear and Chebyshev normshttps://zbmath.org/1517.901262023-09-22T14:21:46.120933Z"Pham, Van Huy"https://zbmath.org/authors/?q=ai:pham.van-huy"Nguyen, Kien Trung"https://zbmath.org/authors/?q=ai:nguyen.kien-trungThe authors consider the inverse 1-median problem on trees with vertex partitions into groups and with variable vertex weights. The cost and elements in each group can be measured by the Chebyshev or a rectilinear norm.
Under the sum of max objective, the modification in each vertex group is measured by the Chebyshev norm and the cost of groups is measured by a rectilinear norm. The authors formulate the problem as a parameterized continuous knapsack. The optimality criterion is derived and a linearithmic algorithm is developed.
Under the max of sum objective, the modification in each vertex group is measured by a rectilinear norm and the cost of groups is measured by the Chebyshev norm. The authors investigate the structure of the corresponding continuous knapsack problem and propose a linearithmic algorithm. They also develop a linear time algorithm based on the special structure of the problem.
Reviewer: Svetlana A. Kravchenko (Minsk)\textsc{influence}: a partizan scoring game on graphshttps://zbmath.org/1517.910042023-09-22T14:21:46.120933Z"Duchêne, Eric"https://zbmath.org/authors/?q=ai:duchene.eric"Gonzalez, Stéphane"https://zbmath.org/authors/?q=ai:gonzalez.stephane"Parreau, Aline"https://zbmath.org/authors/?q=ai:parreau.aline"Rémila, Eric"https://zbmath.org/authors/?q=ai:remila.eric"Solal, Philippe"https://zbmath.org/authors/?q=ai:solal.philippeSummary: We introduce the game \textsc{influence}, a scoring combinatorial game, played on a directed graph where each vertex is either colored black or white. The two players, Black and White, play alternately by taking a vertex of their color and all its successors (for Black) or all its predecessors (for White). The score of each player is the number of vertices he has taken. We prove that \textsc{influence} is a nonzugzwang game, meaning that no player has interest to pass at any step of the game, and thus belongs to Milnor's universe. We study this game in the particular class of paths where black and white vertices are alternated. We give an almost tight strategy for both players when there is one path. More precisely, we prove that the first player always gets a strictly better score than the second one, but that the difference between the scores is bounded by 5. Finally, we exhibit some graphs for which the initial proportion of vertices of the color of a player is as small as possible but where this player can get almost all the vertices.A network approach to link visibility and urban activity locationhttps://zbmath.org/1517.911682023-09-22T14:21:46.120933Z"Natapov, Asya"https://zbmath.org/authors/?q=ai:natapov.asya"Czamanski, Daniel"https://zbmath.org/authors/?q=ai:czamanski.daniel"Fisher-Gewirtzman, Dafna"https://zbmath.org/authors/?q=ai:fisher-gewirtzman.dafnaSummary: Performance of a range of urban amenities is influenced by their accessibility to pedestrians. Success in attracting pedestrians to a particular location depends on how they project visuospatial information. In this paper, we propose an original method for analysing the visuospatial integration of particular locations within a street network. As a case study we analyse the distribution of one type of urban amenities - food and drink public facilities. We represent them in a form of visibility graph as objects of navigational decisions within the street network. To explore how urban facilities, streets and pedestrian visual cognition are interrelated, we create and compare three cases: a street network visibility graph and two visibility graphs of amenities. The first graph is based on the existing, ``natural'' distribution, while the second is an ``artificial'', fabricated version of the environment, where urban locations are redistributed evenly across the case study. We study the graphs' global network properties by the use of small-world, and scale-free models. Our results demonstrate that views available for an urban traveller in the existing, ``natural'' setting had a particular structure. It is built of numerous weakly connected locations coexisting with a small number of hubs with an exceptionally large number of visual connections. Such organisation of urban visibility shows that visuospatial network shares morphological similarities with other natural networks, suggesting that common organizational principles underlie network structure.Is the urban world small? The evidence for small world structure in urban networkshttps://zbmath.org/1517.911692023-09-22T14:21:46.120933Z"Neal, Zachary"https://zbmath.org/authors/?q=ai:neal.zacharySummary: The initial definition of small world networks triggered a rush among network scientists, working in a variety of fields and with data from many different contexts, to identify and document empirical examples of small world networks. Researchers studying urban networks - networks of cities and networks in cities - have also participated in this exercise, but because their work took place in a variety of disciplines, no definitive answer has emerged to the question: Is the urban world small? I answer this question through a systematic review of the evidence for small world structure in 172 urban networks. I find that although authors often claim urban networks are small world (71.5\%), such claims are rarely grounded in a formal index or guided by a specific decision rule, and may overestimate the ubiquity of this structure. However, existing indices of small worldliness offer promising options for summarizing the extent to which an urban network is small world. I conclude with recommendations that urban network researchers make use of these indices and begin conceptualizing small worldliness as a continuous, rather than binary, characteristic.Time-bounded influence diffusion with incentiveshttps://zbmath.org/1517.911742023-09-22T14:21:46.120933Z"Cordasco, Gennaro"https://zbmath.org/authors/?q=ai:cordasco.gennaro"Gargano, Luisa"https://zbmath.org/authors/?q=ai:gargano.luisa"Peters, Joseph G."https://zbmath.org/authors/?q=ai:peters.joseph-g"Rescigno, Adele A."https://zbmath.org/authors/?q=ai:rescigno.adele-anna"Vaccaro, Ugo"https://zbmath.org/authors/?q=ai:vaccaro.ugoSummary: A widely studied model of influence diffusion in social networks represents the network as a graph \(G=(V,E)\) with an influence threshold \(t(v)\) for each node. Initially the members of an initial set \(S\subseteq V\) are influenced. During each subsequent round, the set of influenced nodes is augmented by including every node \(v\) that has at least \(t\)(\(v\)) previously influenced neighbours. The general problem is to find a small initial set that influences the whole network. In this paper we extend this model by using incentives to reduce the thresholds of some nodes. The goal is to minimize the total of the incentives required to ensure that the process completes within a given number of rounds. The problem is hard to approximate in general networks. We present polynomial-time algorithms for paths, trees, and complete networks.
For the entire collection see [Zbl 1400.68027].Network model with scale-free, high clustering coefficients, and small-world propertieshttps://zbmath.org/1517.911832023-09-22T14:21:46.120933Z"Yan, Chuankui"https://zbmath.org/authors/?q=ai:yan.chuankuiSummary: Networks are prevalent in real life, and the study of network evolution models is very important for understanding the nature and laws of real networks. The distribution of the initial degree of nodes in existing classical models is constant or uniform. The model we proposed shows binomial distribution, and it is consistent with real network data. The theoretical analysis shows that the proposed model is scale-free at different probability values and its clustering coefficients are adjustable, and the Barabasi-Albert model is a special case of \(p = 0\) in our model. In addition, the analytical results of the clustering coefficients can be estimated using mean-field theory. The mean clustering coefficients calculated from the simulated data and the analytical results tend to be stable. The model also exhibits small-world characteristics and has good reproducibility for short distances of real networks. Our model combines three network characteristics, scale-free, high clustering coefficients, and small-world characteristics, which is a significant improvement over traditional models with only a single or two characteristics. The theoretical analysis procedure can be used as a theoretical reference for various network models to study the estimation of clustering coefficients. The existence of stable equilibrium points of the model explains the controversy of whether scale-free is universal or not, and this explanation provides a new way of thinking to understand the problem.On M-polynomial and some topological indices of Favipiravir (T-705) and Ribavirinhttps://zbmath.org/1517.920502023-09-22T14:21:46.120933Z"Bharali, Ankur"https://zbmath.org/authors/?q=ai:bharali.ankur"Pegu, Aditya"https://zbmath.org/authors/?q=ai:pegu.aditya"Agarwal, Priyanka"https://zbmath.org/authors/?q=ai:agarwal.priyanka"Chamua, Monjit"https://zbmath.org/authors/?q=ai:chamua.monjitThe paper discusses topological indices, which are numerical quantities used to describe properties of molecular graphs. These graphs represent the carbon-atom skeleton of organic hydrocarbon molecules. Topological indices provide a way to numerically characterize the structure and properties of these molecules.
For a graph \(G=(V,E)\) and a function \(f\), a degree-based topological index is a graph invariant given by the formula: \[I_f(G)= \sum_{uv \in E(G)} f(d_u,d_v),\] where \(d_u\) denotes the degree of a vertex \(u\) in \(G\).
To determine certain distance-based topological indices, algebraic polynomials can be used. One important example is the M-polynomial introduced by \textit{E. Deutsch} and \textit{S. Klavžar} [Iran. J. Math. Chem. 6, No. 2, 93--102 (2015; Zbl 1367.05048)], defined as: \[M(G; x ,y)= \sum_{i \le j} m_{i,j}(G) x^i y^j,\] where \(m_{i,j}(G)\) represents the number of edges \(uv \in E(G)\) such that \(\{d_u,d_v\} = \{i,j\}\).
This M-polynomial allows for the calculation of closed-form expressions for many degree-based topological indices by using certain derivatives or integrals of this polynomial.
The authors of the paper focus on determining the M-polynomial of two antiviral compounds: Favipiravir and Ribavirin. Using the established results, they deduced some degree-based topological indices for both molecules. Specifically, for Favipiravir and Ribavirin they computed the hyper-Zagreb index, reduced reciprocal Randić index and atom-bond connectivity index. Additionally, they also calculated the forgotten topological index specifically for Ribavirin.
Reviewer: Aleksander Vesel (Maribor)Tile-based modeling of DNA self-assembly for two graph families with appended pathshttps://zbmath.org/1517.920512023-09-22T14:21:46.120933Z"Griffin, Chloe"https://zbmath.org/authors/?q=ai:griffin.chloe"Sorrells, Jessica"https://zbmath.org/authors/?q=ai:sorrells.jessicaIn 1974, twenty one years after the famous discovery of the structure of the DNA by Watson and Crick, N. Taniguchi, professor of the Tokyo Science University, introduced the term ``nanotechnology'' [\textit{N. Taniguchi}, ``On the basic concept of `nano-technology''', in: Proceedings of the International Conference on Production Engineering. 26--29 (1974)] for describing the engineering of materials at an atomic level. Three years later, at another corner of the Earth, Nadrian (Ned) C. Seeman (1945--2021) was appointed, after earning his PhD in biochemistry at the University of Chicago followed by postdoctoral work first at Columbia University and, then, with the legendary crystallographer A. Rich at the M.I.T. on the structure of large biological molecules. Istvan and Balazs Hargittai dubbed him the pioneer of DNA nanotechnology: [\textit{I. Hargittai} and \textit{B. Hargittai}, Struct. Chem. 33, 631--633 (2022; \url{doi:10.1007/s11224-022-01894-3})]. While not especially successful in various Projects there, Seeman stumbled upon the concept of the so called Holliday junction which caught his imagination. It is worth saying, though, that Seeman's motto was next (1995) ``Just because we can't do it doesn't mean we can't think about doing it''. Recall that Holliday junction was described by the molecular biologist Robin Holliday in 1964. It is a branched nucleic acid structure that joins four double stranded arms. Playing with its model led Seeman to develop his life motif, constructing three-dimensional DNA structures, and he became a nano-architect, inventing DNA nanotechnology in an attempt to better the crystallization process [\textit{E. Winfree} et al., Nature 394, 539--544 (1998; \url{doi:10.1038/28998})]. A key technology of DNA nanotechnology became in the early 2000s DNA-nanostructure self-assembling of some specially designed DNA strands -- the so called target structures. The latter can mathematically be represented as discrete graphs. In this case one can use the Ellis-Monaghan model of a k-armed branched junction molecule that is a molecule of DNA consisting of a center point with extending arms (see Figure 1).
Self-assembling of branched molecules of DNA into nanostructures via a complementary cohesive base-pairing is the theme of the present work under review. Its aim is to minimize the number of different types of branched junction molecules necessary to assemble certain target structures. For this purpose this work represents target structures as discrete graphs and branched DNA molecules as vertices with half-edges and presents the minimum numbers of required branched molecule and cohesive-end types under three levels of restrictive conditions for the tadpole and lollipop graph families which represent cycle and complete graphs with a path appended via a single cut-vertex. Three general lemmas regarding these vertex-induced path subgraphs are derived. A summary of the results of three different theoretical lab scenarios is given in Table 3. These results may be helpful in future studies on graphs with path subgraphs appended via a single cut-vertex.
Reviewer: Eugene Kryachko (Kyjiw)Adventures in supersingularlandhttps://zbmath.org/1517.940572023-09-22T14:21:46.120933Z"Arpin, Sarah"https://zbmath.org/authors/?q=ai:arpin.sarah"Camacho-Navarro, Catalina"https://zbmath.org/authors/?q=ai:camacho-navarro.catalina"Lauter, Kristin"https://zbmath.org/authors/?q=ai:lauter.kristin-e"Lim, Joelle"https://zbmath.org/authors/?q=ai:lim.joelle"Nelson, Kristina"https://zbmath.org/authors/?q=ai:nelson.kristina"Scholl, Travis"https://zbmath.org/authors/?q=ai:scholl.travis"Sotáková, Jana"https://zbmath.org/authors/?q=ai:sotakova.janaSummary: Supersingular Isogeny Graphs were introduced as a source of hard problems in cryptography by \textit{D. X. Charles} et al. [J. Cryptology 22, No. 1, 93--113 (2009; Zbl 1166.94006)] for the construction of cryptographic hash functions and have been used for key exchange SIKE. The security of such systems depends on the difficulty of finding a path between two random vertices. In this article, we study several aspects of the structure of these graphs. First, we study the subgraph given by \(j\)-invariants in \(\mathbb{F}_p\), using the related isogeny graph consisting of only \(\mathbb{F}_p\)-rational curves and isogenies. We prove theorems on how the connected components thereof attach, stack, and fold when mapped into the full graph. The \(\mathbb{F}_p\)-rational vertices are fixed by the Frobenius involution on the graph, and we call the induced graph the spine. Finding paths to the spine is relevant in cryptanalysis. Second, we present numerous computational experiments and heuristics relating to the position of the spine within the whole graph. These include studying the distance of random vertices to the spine, estimates of the diameter of the graph, how often paths are preserved under the Frobenius involution, and what proportion of vertices are conjugate. We compare some of the heuristics with known results on other Ramanujan graphs.Provably secure reflection ciphershttps://zbmath.org/1517.940632023-09-22T14:21:46.120933Z"Beyne, Tim"https://zbmath.org/authors/?q=ai:beyne.tim"Chen, Yu Long"https://zbmath.org/authors/?q=ai:chen.yulongSummary: This paper provides the first analysis of reflection ciphers such as \textsc{Prince} from a provable security viewpoint.
As a first contribution, we initiate the study of key-alternating reflection ciphers in the ideal permutation model. Specifically, we prove the security of the two-round case and give matching attacks. The resulting security bound takes form \(\mathcal{O}(qp^2/2^{2n}+q^2/2^n)\), where \(q\) is the number of construction evaluations and \(p\) is the number of direct adversarial queries to the underlying permutation. Since the two-round construction already achieves an interesting security lower bound, this result can also be of interest for the construction of reflection ciphers based on a single public permutation.
Our second contribution is a generic key-length extension method for reflection ciphers. It provides an attractive alternative to the FX construction, which is used by \textsc{Prince} and other concrete key-alternating reflection ciphers. We show that our construction leads to better security with minimal changes to existing designs. The security proof is in the ideal cipher model and relies on a reduction to the two-round Even-Mansour cipher with a single round key. In order to obtain the desired result, we sharpen the bad-transcript analysis and consequently improve the best-known bounds for the single-key Even-Mansour cipher with two rounds. This improvement is enabled by a new sum-capture theorem that is of independent interest.
For the entire collection see [Zbl 1514.94004].\(P_4\)-free partition and cover numbers \& applicationshttps://zbmath.org/1517.940662023-09-22T14:21:46.120933Z"Block, Alexander R."https://zbmath.org/authors/?q=ai:block.alexander-r"Brânzei, Simina"https://zbmath.org/authors/?q=ai:branzei.simina"Maji, Hemanta K."https://zbmath.org/authors/?q=ai:maji.hemanta-k"Mehta, Himanshi"https://zbmath.org/authors/?q=ai:mehta.himanshi"Mukherjee, Tamalika"https://zbmath.org/authors/?q=ai:mukherjee.tamalika"Nguyen, Hai H."https://zbmath.org/authors/?q=ai:nguyen.hai-hSummary: \(P_4\)-free graphs also known as cographs, complement-reducible graphs, or hereditary Dacey graphs have been well studied in graph theory. Motivated by computer science and information theory applications, our work encodes (flat) joint probability distributions and Boolean functions as bipartite graphs and studies bipartite \(P_4\)-free graphs. For these applications, the graph properties of edge partitioning and covering a bipartite graph using the minimum number of these graphs are particularly relevant. Previously, such graph properties have appeared in leakage-resilient cryptography and (variants of) coloring problems. Interestingly, our covering problem is closely related to the well-studied problem of product (a.k.a., Prague) dimension of loopless undirected graphs, which allows us to employ algebraic lower-bounding techniques for the product/Prague dimension. We prove that computing these numbers is NP-complete, even for bipartite graphs. We establish a connection to the (unsolved) Zarankiewicz problem to show that there are bipartite graphs with size-\(N\) partite sets such that these numbers are at least \(\varepsilon\cdot N^{1-2 \varepsilon}\), for \(\varepsilon\in \{1/3,1/4,1/5,\dots\}\). Finally, we accurately estimate these numbers for bipartite graphs encoding well-studied Boolean functions from circuit complexity, such as set intersection, set disjointness, and inequality. For applications in information theory and communication \& cryptographic complexity, we consider a system where a setup samples from a (flat) joint distribution and gives the participants, Alice and Bob, their portion from this joint sample. Alice and Bob's objective is to non-interactively establish a shared key and extract the left-over entropy from their portion of the samples as independent private randomness. A genie, who observes the joint sample, provides appropriate assistance to help Alice and Bob with their objective. Lower bounds to the minimum size of the genie's assistance translate into communication and cryptographic lower bounds. We show that (the \(\log_2\) of) the \(P_4\)-free partition number of a graph encoding the joint distribution that the setup uses is equivalent to the size of the genie's assistance. Consequently, the joint distributions corresponding to the bipartite graphs constructed above with high \(P_4\)-free partition numbers correspond to joint distributions requiring more assistance from the genie. As a representative application in non-deterministic communication complexity, we study the communication complexity of nondeterministic protocols augmented by access to the equality oracle at the output. We show that (the \(\log_2\) of) the \(P_4\)-free cover number of the bipartite graph encoding a Boolean function \(f\) is equivalent to the minimum size of the nondeterministic input required by the parties (referred to as the communication complexity of \(f\) in this model). Consequently, the functions corresponding to the bipartite graphs with high \(P_4\)-free cover numbers have high communication complexity. Furthermore, there are functions with communication complexity close to the naïve protocol where the nondeterministic input reveals a party's input. Finally, the access to the equality oracle reduces the communication complexity of computing set disjointness by a constant factor in contrast to the model where parties do not have access to the equality oracle. To compute the inequality function, we show an exponential reduction in the communication complexity, and this bound is optimal. On the other hand, access to the equality oracle is (nearly) useless for computing set intersection.
For the entire collection see [Zbl 1465.94005].Must the communication graph of MPC protocols be an expander?https://zbmath.org/1517.940722023-09-22T14:21:46.120933Z"Boyle, Elette"https://zbmath.org/authors/?q=ai:boyle.elette"Cohen, Ran"https://zbmath.org/authors/?q=ai:cohen.ran"Data, Deepesh"https://zbmath.org/authors/?q=ai:data.deepesh"Hubáček, Pavel"https://zbmath.org/authors/?q=ai:hubacek.pavelSummary: Secure multiparty computation (MPC) on incomplete communication networks has been studied within two primary models: (1) where a partial network is fixed a priori, and thus corruptions can occur dependent on its structure, and (2) where edges in the communication graph are determined dynamically as part of the protocol. Whereas a rich literature has succeeded in mapping out the feasibility and limitations of graph structures supporting secure computation in the fixed-graph model (including strong classical lower bounds), these bounds do not apply in the latter dynamic-graph setting, which has recently seen exciting new results, but remains relatively unexplored. In this work, we initiate a similar foundational study of MPC within the dynamic-graph model. As a first step, we investigate the property of graph expansion. All existing protocols (implicitly or explicitly) yield communication graphs which are expanders, but it is not clear whether this is inherent. Our results consist of two types (for constant fraction of corruptions):
\begin{itemize}
\item Upper bounds: We demonstrate secure protocols whose induced communication graphs are not expander graphs, within a wide range of settings (computational, information theoretic, with low locality, even with low locality and adaptive security), each assuming some form of input-independent setup.
\item Lower bounds: In the plain model (no setup) with adaptive corruptions, we demonstrate that for certain functionalities, no protocol can maintain a non-expanding communication graph against all adversarial strategies. Our lower bound relies only on protocol correctness (not privacy) and requires a surprisingly delicate argument.
\end{itemize}
More generally, we provide a formal framework for analyzing the evolving communication graph of MPC protocols, giving a starting point for studying the relation between secure computation and further, more general graph properties.Correlated pseudorandomness from expand-accumulate codeshttps://zbmath.org/1517.940732023-09-22T14:21:46.120933Z"Boyle, Elette"https://zbmath.org/authors/?q=ai:boyle.elette"Couteau, Geoffroy"https://zbmath.org/authors/?q=ai:couteau.geoffroy"Gilboa, Niv"https://zbmath.org/authors/?q=ai:gilboa.niv"Ishai, Yuval"https://zbmath.org/authors/?q=ai:ishai.yuval"Kohl, Lisa"https://zbmath.org/authors/?q=ai:kohl.lisa"Resch, Nicolas"https://zbmath.org/authors/?q=ai:resch.nicolas"Scholl, Peter"https://zbmath.org/authors/?q=ai:scholl.peterSummary: A pseudorandom correlation generator (PCG) is a recent tool for securely generating useful sources of correlated randomness, such as random oblivious transfers (OT) and vector oblivious linear evaluations (VOLE), with low communication cost.
We introduce a simple new design for PCGs based on so-called expand-accumulate codes, which first apply a sparse random expander graph to replicate each message entry, and then accumulate the entries by computing the sum of each prefix. Our design offers the following advantages compared to state-of-the-art PCG constructions:
\begin{itemize}
\item Competitive concrete efficiency backed by provable security against relevant classes of attacks;
\item An offline-online mode that combines near-optimal cache-friendliness with simple parallelization;
\item Concretely efficient extensions to pseudorandom correlation \textit{functions}, which enable incremental generation of new correlation instances on demand, and to new kinds of correlated randomness that include circuit-dependent correlations.
\end{itemize}
To further improve the concrete computational cost, we propose a method for speeding up a full-domain evaluation of a puncturable pseudorandom function (PPRF). This is independently motivated by other cryptographic applications of PPRFs.
For the entire collection see [Zbl 1514.94002].Simon's algorithm and symmetric crypto: generalizations and automatized applicationshttps://zbmath.org/1517.940762023-09-22T14:21:46.120933Z"Canale, Federico"https://zbmath.org/authors/?q=ai:canale.federico"Leander, Gregor"https://zbmath.org/authors/?q=ai:leander.gregor"Stennes, Lukas"https://zbmath.org/authors/?q=ai:stennes.lukasSummary: In this paper we deepen our understanding of how to apply Simon's algorithm to break symmetric cryptographic primitives.
On the one hand, we automate the search for new attacks. Using this approach we automatically find the first efficient key-recovery attacks against constructions like 5-round MISTY L-FK or 5-round Feistel-FK (with internal permutation) using Simon's algorithm.
On the other hand, we study generalizations of Simon's algorithm using non-standard Hadamard matrices, with the aim to expand the quantum symmetric cryptanalysis toolkit with properties other than the periods. Our main conclusion here is that none of these generalizations can accomplish that, and we conclude that exploiting non-standard Hadamard matrices with quantum computers to break symmetric primitives will require fundamentally new attacks.
For the entire collection see [Zbl 1514.94003].Designing tweakable enciphering schemes using public permutationshttps://zbmath.org/1517.940782023-09-22T14:21:46.120933Z"Chakraborty, Debrup"https://zbmath.org/authors/?q=ai:chakraborty.debrup"Dutta, Avijit"https://zbmath.org/authors/?q=ai:dutta.avijit"Kundu, Samir"https://zbmath.org/authors/?q=ai:kundu.samirSummary: A tweakable enciphering scheme (TES) is a length preserving (tweakable) encryption scheme that provides (tweakable) strong pseudorandom permutation security on arbitrarily long messages. TES is traditionally built using block ciphers and the security of the mode depends on the strong pseudorandom permutation security of the underlying block cipher. In this paper, we construct TESs using public random permutations. Public random permutations are being considered as a replacement of block cipher in several cryptographic schemes including AEs, MACs, etc. However, to our knowledge, a systematic study of constructing TES using public random permutations is missing. In this paper, we give a generic construction of a TES which uses a public random permutation, a length expanding public permutation based PRF and a hash function which is both almost xor universal and almost regular. Further, we propose a concrete length expanding public permutation based PRF construction. We also propose a single keyed TES using a public random permutation and an AXU and almost regular hash function.Simplified MITM modeling for permutations: new (quantum) attackshttps://zbmath.org/1517.941522023-09-22T14:21:46.120933Z"Schrottenloher, André"https://zbmath.org/authors/?q=ai:schrottenloher.andre"Stevens, Marc"https://zbmath.org/authors/?q=ai:stevens.marcSummary: Meet-in-the-middle (MITM) is a general paradigm where internal states are computed along two independent paths (`forwards' and `backwards') that are then matched. Over time, MITM attacks improved using more refined techniques and exploiting additional freedoms and structure, which makes it more involved to find and optimize such attacks. This has led to the use of detailed attack models for generic solvers to automatically search for improved attacks, notably a MILP model developed by \textit{Z. Bao} et al. [Lect. Notes Comput. Sci. 12696, 771--804 (2021; Zbl 1479.94121)].
In this paper, we study a simpler MILP modeling combining a greatly reduced attack representation as input to the generic solver, together with a theoretical analysis that, for any solution, proves the existence and complexity of a detailed attack. This modeling allows to find both classical and quantum attacks on a broad class of cryptographic permutations. First, Present-like constructions, with the permutations from the Spongent hash functions: we improve the MITM step in distinguishers by up to 3 rounds. Second, AES-like designs: despite being much simpler than Bao et al.'s [loc. cit.], our model allows to recover the best previous results. The only limitation is that we do not use degrees of freedom from the key schedule. Third, we show that the model can be extended to target more permutations, like Feistel networks. In this context we give new Guess-and-determine attacks on reduced \textsf{Simpira v2} and \textsc{Sparkle}.
Finally, using our model, we find several new quantum preimage and pseudo-preimage attacks (e.g. \textsf{Haraka v2}, \textsf{Simpira v2}\dots) targeting the same number of rounds as the classical attacks.
For the entire collection see [Zbl 1514.94003].On ideal and weakly-ideal access structureshttps://zbmath.org/1517.941992023-09-22T14:21:46.120933Z"Kaboli, Reza"https://zbmath.org/authors/?q=ai:kaboli.reza"Khazaei, Shahram"https://zbmath.org/authors/?q=ai:khazaei.shahram"Parviz, Maghsoud"https://zbmath.org/authors/?q=ai:parviz.maghsoudSummary: For more than two decades, proving or refuting the following statement has remained a challenging open problem in the theory of secret sharing schemes (SSSs): every ideal access structure admits an ideal perfect multi-linear SSS. The class of group-characterizable (GC) SSSs include the multi-linear ones. Hence, if the above statement is true, then so is the following weaker statement: every ideal access structure admits an ideal perfect GC SSS. One contribution of this paper is to show that ideal SSSs are not necessarily GC. Our second contribution is to study the above two statements with respect to several variations of weakly-ideal access structures. Recently, \textit{C. Mejia} and \textit{J. A. Montoya} [J. Inf. Optim. Sci. 39, No. 7, 1463--1482 (2018; \url{doi.org/10.1080/02522667.2017.1367513})] studied ideal access structures that admit ideal multi-linear schemes and provided a classification-like theorem for them. We additionally present some tools that are useful to extend their result.Extremal set theory and LWE based access structure hiding verifiable secret sharing with malicious-majority and free verificationhttps://zbmath.org/1517.942052023-09-22T14:21:46.120933Z"Sehrawat, Vipin Singh"https://zbmath.org/authors/?q=ai:sehrawat.vipin-singh"Yeo, Foo Yee"https://zbmath.org/authors/?q=ai:yeo.foo-yee"Desmedt, Yvo"https://zbmath.org/authors/?q=ai:desmedt.yvo-gSummary: Secret sharing allows a dealer to distribute a secret among a set of parties such that only authorized subsets, specified by an access structure, can reconstruct the secret. \textit{V. S. Sehrawat} and \textit{Y. Desmedt} [Lect. Notes Comput. Sci. 12273, 246--261 (2020; Zbl 07336109)] introduced hidden access structures , that remain secret until some authorized subset of parties collaborate. However, their scheme assumes semi-honest parties and supports only restricted access structures. We address these shortcomings by constructing a novel access structure hiding verifiable secret sharing scheme that supports all monotone access structures. Our scheme is the first secret sharing solution to support malicious behavior identification and share verifiability in malicious-majority settings. Furthermore, the verification procedure of our scheme incurs no communication overhead, and is therefore ``free''. As the building blocks of our scheme, we introduce and construct the following:
\begin{itemize}
\item a set-system with greater than \(\exp\left(c\frac{2(\log h)^2}{(\log\log h)}\right)+2\exp\left(c\frac{(\log h)^2}{(\log\log h)}\right)\) subsets of a set of \(h\) elements. Our set-system, \(\mathcal{H}\), is defined over \(\mathbb{Z}_m\), where \(m\) is a non-prime-power. The size of each set in \(\mathcal{H}\) is divisible by \(m\) while the sizes of the pairwise intersections of different sets are not divisible by \(m\) unless one set is a (proper) subset of the other,
\item a new variant of the learning with errors (LWE) problem, called \textsf{PRIM-LWE}, wherein the secret matrix is sampled such that its determinant is a generator of \(\mathbb{Z}_q^\ast \), where \(q\) is the LWE modulus.
\end{itemize}
Our scheme arranges parties as nodes of a directed acyclic graph and employs modulus switching during share generation and secret reconstruction. For a setting with \(\mathfrak{l}\) parties, our (non-linear) scheme supports all \(2^{2^{\ell-O(\log \ell)}}\) monotone access structures, and its security relies on the hardness of the LWE problem. Our scheme's maximum share size, for any access structure, is:
\[
(1+o(1))\frac{ 2^\ell}{\sqrt{\pi\ell /2}}(2q^{\varrho+0.5}+\sqrt{q}+\Theta(h)),
\]
where \(\varrho\leq 1\) is a constant. We provide directions for future work to reduce the maximum share size to:
\[
\frac{1}{l+1}\left((1+o(1))\frac{2^\ell}{\sqrt{\pi\ell/2}}(2q^{\varrho+0.5}+2\sqrt{q})\right),
\]
where \(l\geq 2\). We also discuss three applications of our secret sharing scheme.