×

zbMATH — the first resource for mathematics

Burr, Michael A.; Lipman, Drew J.
Quadratic-monomial generated domains from mixed signed, directed graphs. (English) Zbl 1414.05307
Int. J. Algebra Comput. 29, No. 2, 279-308 (2019).
MSC:
05E15 Combinatorial aspects of groups and algebras (MSC2010)
05C20 Directed graphs (digraphs), tournaments
05C22 Signed and weighted graphs
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
13G05 Integral domains
13C14 Cohen-Macaulay modules
Keywords:
edge rings; odd cycle condition; normal domains; normalization; Serre’s \(R_1\) condition; Cohen-Macaulay rings; quadratic-monomial generated domains
Software:
Normaliz
PDF BibTeX XML Cite
\textit{M. A. Burr} and \textit{D. J. Lipman}, Int. J. Algebra Comput. 29, No. 2, 279--308 (2019; Zbl 1414.05307)
Full Text: DOI
References:
[1] Bermejo, I.; García Marco, I.; Reyes, E., Graphs and complete intersection toric ideals, J. Algebra Appl., 14, 9, 37, (2015) · Zbl 1333.14046
[2] Bermejo, I.; Gimenez, P.; Simis, A., Polar syzygies in characteristic zero: The monomial case, J. Pure Appl. Algebra, 213, 1, 1-21, (2009) · Zbl 1179.13007
[3] Bruns, W.; Herzog, J., Cohen-Macaulay Rings, (1996), Cambridge University Press
[4] Bruns, W.; Vasconcelos, W. V.; Villarreal, R. H., Degree bounds in monomial subrings, Illinois J. Math., 41, 3, 341-353, (1997) · Zbl 0908.13014
[5] Cox, D.; Little, J.; Schenck, H., Toric Varieties, (2011), American Mathematical Society · Zbl 1223.14001
[6] D’Alì, A., Toric ideals associated with gap-free graphs, J. Pure Appl. Algebra, 219, 9, 3862-3872, (2015) · Zbl 1435.13022
[7] Estrada, M. E., On a theorem of Fröberg and saturated graphs, Rev. Colomb. Mat., 34, 1, 19-24, (2000) · Zbl 1015.13006
[8] Gitler, I.; Valencia, C. E., Bounds for invariants of edge-rings, Comm. Algebra, 33, 5, 1603-1616, (2005) · Zbl 1077.13012
[9] Gitler, I.; Villarreal, R. H., Aportaciones Matemáticas: Textos, Graphs, rings and polyhedra, (2011), Instituto de Matemáticas: Instituto de Matemáticas, UNAM, México
[10] Grothendieck, A., Éléments de géométrie algébrique. IV. étude locale des schémas et des morphismes de schémas. II, Inst. Haut. Ét. Sci. Publ. Math., 24, 231, (1965) · Zbl 0135.39701
[11] Herzog, J.; Hibi, T.; Zheng, X., Monomial ideals whose powers have a linear resolution, Math. Scand., 95, 1, 23-32, (2004) · Zbl 1091.13013
[12] Hibi, T.; Higashitani, A.; Kimura, K.; O’Keefe, A. B., Depth of initial ideals of normal edge rings, Commun. Algebra, 42, 7, 2908-2922, (2014) · Zbl 1297.13036
[13] Hibi, T.; Katthän, L., Edge rings satisfying Serre’s condition (\(R_1\)), Proc. Amer. Math. Soc., 142, 7, 2537-2541, (2014) · Zbl 1306.13015
[14] Hibi, T.; Matsuda, K.; Ohsugi, H., Strongly Koszul edge rings, Acta Math. Viet., 41, 1, 69-76, (2016) · Zbl 1335.13019
[15] Hibi, T.; Mori, A.; Ohsugi, H.; Shikama, A., The number of edges of the edge polytope of a finite simple graph, Ars Math. Contem., 10, 2, 323-332, (2016) · Zbl 1351.52007
[16] Hibi, T.; Nishiyama, K.; Ohsugi, H.; Shikama, A., Many toric ideals generated by quadratic binomials possess no quadratic Gröbner bases, J. Algebra, 408, 138-146, (2014) · Zbl 1304.13040
[17] Huneke, C.; Swanson, I., Integral Closure of Ideals, Rings, and Modules, 336, (2006), Cambridge University Press: Cambridge University Press, Cambridge · Zbl 1117.13001
[18] D. Lipman and M. Burr, Serre’s properties for quadratic generated domains from graphs, Technical Report, arXiv:1708.05809 [math.AG] (2017).
[19] Matsui, T., Ehrhart series for connected simple graphs, Graphs Combin., 29, 3, 617-635, (2013) · Zbl 1267.05026
[20] Ohsugi, H.; Hibi, T., Normal polytopes arising from finite graphs, J. Algebra, 207, 2, 409-426, (1998) · Zbl 0926.52017
[21] Ohsugi, H.; Hibi, T., Compressed polytopes, initial ideals and complete multipartite graphs, Illinois J. Math., 44, 2, 391-406, (2000) · Zbl 0943.13016
[22] Ohsugi, H.; Hibi, T., Special simplices and Gorenstein toric rings, J. Combin. Theory. Ser. A, 113, 4, 718-725, (2006) · Zbl 1095.13023
[23] Ohsugi, H.; Hibi, T., A Gröbner basis characterization for chordal comparability graphs, European J. Combin., 59, 122-128, (2017) · Zbl 1348.05098
[24] Serre, J.-P., Local Algebra, (2000), Springer-Verlag: Springer-Verlag, Berlin
[25] Simis, A.; Vasconcelos, W. V.; Villarreal, R. H., On the ideal theory of graphs, J. Algebra, 167, 2, 389-416, (1994) · Zbl 0816.13003
[26] Simis, A.; Vasconcelos, W. V.; Villarreal, R. H., The integral closure of subrings associated to graphs, J. Algebra, 199, 1, 281-289, (1998) · Zbl 0902.13004
[27] Sullivant, S., Combinatorial symbolic powers, J. Algebra, 319, 1, 115-142, (2008) · Zbl 1133.13027
[28] Tatakis, C.; Thoma, A., On complete intersection toric ideals of graphs, J. Algeb. Combin. Int. J., 38, 2, 351-370, (2013) · Zbl 1282.14089
[29] Vasconcelos, W., Integral Closure, (2005), Springer-Verlag: Springer-Verlag, Berlin
[30] Villarreal, R. H., Rees algebras of edge ideals, Commun. Algebra, 23, 9, 3513-3524, (1995) · Zbl 0836.13014
[31] Villarreal, R. H., Normality of subrings generated by square free monomials, J. Pure Appl. Algebra, 113, 1, 91-106, (1996) · Zbl 0862.13008
[32] Villarreal, R. H., Normality of semigroups with some links to graph theory, Discr. Math., 302, 1-3, 267-284, (2005) · Zbl 1103.20058
[33] Vitulli, M. A., Serre’s condition \(R_\ell\) for affine semigroup rings, Commun. Algebra, 37, 3, 743-756, (2009) · Zbl 1170.13002
[34] Bruns, W.; Koch, R., Computing the integral closure of an affine semigroup, Univer. Iagell. Acta Math., 39, 59-70, (2001) · Zbl 1006.20045
[35] Zaslavsky, T., Totally frustrated states in the chromatic theory of gain graphs, European J. Combin., 30, 1, 133-156, (2009) · Zbl 1165.05011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.