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The \(v\)-number of edge ideals. (English) Zbl 1448.13037

Let \(X=\{P_1,\dots,P_m\}\subset \mathbb{P}^{n-1}\) be a set of \(m\) points in projective space over \(K\). In [A. V. Geramita et al., Trans. Am. Math. Soc. 339, No. 1, 163–189 (1993; Zbl 0793.14002)] the degree of \(P_i\) in \(X\) is defined to be the smallest \(d\geq 1\) such that there exists a hypersurface of degree \(d\) passing through all of the points of \(X\) except for \(P_i\). If \(I\subset S = K[x_1,\dots,x_n]\) is the homogeneous ideal of \(X\), the degree of \(P_i\) in \(X\) is thus \[ \min \{d\geq 1 : \exists\, f \in S_d \text{ with } (I:f) = \mathfrak{p}_i\}, \] where \(\mathfrak{p}_i\) is the homogeneous ideal of \(\{P_i\}\). In [S. M. Cooper et al., Adv. Appl. Math. 112, Article ID 101940, 34 p. (2020; Zbl 1428.13048)] a generalization of this number is defined for any homogeneous ideal \(I\subset S\) different from the irrelevant ideal with respect to each one of its associated primes \(\mathfrak{p}_1,\dots,\mathfrak{p}_m\), using the formula above. If \(\mathrm{v}_{\mathfrak{p}_i}(I)\) denotes this number, then the \(\mathrm{v}\)-number of \(I\), is defined in [Cooper et al., loc. cit.] by \(\min\{\mathrm{v}_{\mathfrak{p}_1}(I),\dots,\mathrm{v}_{\mathfrak{p}_m}(I)\}\).
The article under review contains a study of the \(\mathrm{v}\)-number of edge ideals of clutters or, equivalently, square-free monomial ideals. If \(I\) is not prime the authors show that the \(\mathrm{v}\)-number is equal to the minimum cardinality of a stable set of the associated clutter, whose neighbor set is a minimal vertex cover (Theorem 3.5). Following this, the authors look into homological invariants of \(I\) and their relation with the \(\mathrm{v}\)-number. They show that if \(I\) is the Stanley-Reisner ideal of a vertex decomposable simplicial complex then the \(\mathrm{v}\)-number of \(I\) is a lower bound for the Castelnuovo-Mumford regularity of the quotient graded ring \(S/I\) (Theorem 3.13). They show that the regularity of the quotient graded ring is equal to \(\mathrm{v}(I)\) when \(I\) is the cover ideal of a graph whose independence complex is pure and shellable (Propositionn 3.17), or when \(I\) is the edge ideal of graph such that \(I\) has a linear resolution (Proposition 3.18). In the end of Section 3 it is shown that the \(\mathrm{v}\)-number of the edge ideal of the whisker graph is the independent domination number of the (original) graph, i.e., the minimum cardinality of a maximal stable set of vertices (Theorem 3.19).
In the fourth part of the article, the authors use the \(\mathrm{v}\)-number to study \(W_2\)-graphs. (A graph is in the class of \(W_2\) if and only if there are at least two vertices and any two disjoint disjoint stable sets are contained in two maximal disjoint stable sets.) They show that a graph \(G\) is in \(W_2\) if and only if \(G\) is well covered (every maximal stable set has the maximum cardinality of a stable set) and the family of maximal stable sets of vertices coincides with the family of stable sets of vertices whose neighbor set is a minimal vertex cover if and only if \(\mathrm{v}(I)=\dim S/I\), where \(I\) is the edge ideal of the graph (Theorems 4.3 and 4.5). In connection with the class of \(W_2\)-graphs, the authors now turn their attention to edge critical graphs. Combining their results with a result of [D. T. Hoang et al., J. Comb. Theory, Ser. A 120, No. 5, 1073–1086 (2013; Zbl 1277.05174)] they show that if the second symbolic power of the the edge ideal of a graph is Cohen-Macaulay then the graph is edge critical (Theorem 4.7). For graphs with independence number equal to two, they extend this to an equivalence (Theorem 4.18).
In the final section of this article the authors apply Theorem 4.7 in a Macaulay2 procedure to produce lists of edge critical graphs.

MSC:

13F20 Polynomial rings and ideals; rings of integer-valued polynomials
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
05E40 Combinatorial aspects of commutative algebra

Software:

Macaulay2
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References:

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