×

zbMATH — the first resource for mathematics

Matching numbers and the regularity of the Rees algebra of an edge ideal. (English) Zbl 1446.13018
Let \(G\) be a finite simple graph with vertex set \(\{1,\dots,n\}\), without isolated vertices. Denote its edge set by \(E(G)\). The edge ideal is the ideal of the polynomial ring \(K[x_1,\dots,x_n]\) given by \(I=(x_ix_j : \{i,j\}\in E_G)\). In the work under review, the authors study the Castelnuovo-Mumford regularity of the Rees algebra of \(I\), which is defined by \(R(I)=\bigoplus_{s\geq 0} I^s\). In the main result (Theorem 2.2) they show that under the assumptions that \(|E(G)|\geq 2\) and that \(R(I)\) is normal, \[ \operatorname{mat}(G) \leq \operatorname{reg} R(I) \leq \operatorname{mat}(G)+1, \] where \(\operatorname{mat}(G)\) denotes the the maximum cardinality of a matching of \(G\). A corollary of the proof is that when, in addition to the above assumptions, \(G\) has a perfect matching, \(\operatorname{reg} R(I) = \operatorname{mat}(G)\).
When \(G\) is bipartite, the regularity of the Rees algebra of \(I\) is equal to \(\operatorname{mat}(G)\). This follows from Proposition 4.5 in [I. Gitler et al., J. Pure Appl. Algebra 201, No. 1–3, 17–24 (2005; Zbl 1081.13002)] and is also the statement of Theorem 4.2 in [Y. Cid-Ruiz, Matematiche 73, No. 2, 279–296 (2018; Zbl 1427.13017)].
MSC:
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
13D02 Syzygies, resolutions, complexes and commutative rings
05E40 Combinatorial aspects of commutative algebra
PDF BibTeX Cite
Full Text: DOI
References:
[1] Bruns, W.; Herzog, J., Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics 39 (1993), Cambridge: Cambridge University Press, Cambridge · Zbl 0788.13005
[2] Bruns, W.; Herzog, J., Semigroup rings and simplicial complexes, J. Pure Appl. Algebra, 122, 185-208 (1997) · Zbl 0884.13006
[3] Cid-Ruiz, Y., Regularity and Gröbner bases of the Rees algebra of edge ideals of bipartite graphs, Le Matematiche, 73, 279-296 (2018) · Zbl 1427.13017
[4] Gitler, I.; Valencia, C.; Villarreal, RH, A note on the Rees algebra of a bipartite graph, J. Pure Appl. Algebra, 201, 17-24 (2005) · Zbl 1081.13002
[5] J. Herzog and T. Hibi, Monomial ideals, Graduate Texts in Mathematics 260, Springer, 2011. · Zbl 1206.13001
[6] J. Herzog, T. Hibi and H. Ohsugi, Binomial Ideals, Graduate Texts in Mathematics 279, Springer, 2018. · Zbl 1403.13004
[7] Ohsugi, H.; Hibi, T., Normal polytopes arising from finite graphs, J. Algebra, 207, 409-426 (1998) · Zbl 0926.52017
[8] Stanley, RP, A monotonicity property of \(h\)-vectors and \(h^*\)-vectors, Europ. J. Combin., 14, 251-258 (1993) · Zbl 0799.52008
[9] R.H: Villarreal, Monomial Algebras (Second Edition), Chapman and Hall/CRC, 2015. · Zbl 1325.13004
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.