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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
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##### References:
  Bruns, W.; Herzog, J., Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics 39 (1993), Cambridge: Cambridge University Press, Cambridge · Zbl 0788.13005  Bruns, W.; Herzog, J., Semigroup rings and simplicial complexes, J. Pure Appl. Algebra, 122, 185-208 (1997) · Zbl 0884.13006  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  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  J. Herzog and T. Hibi, Monomial ideals, Graduate Texts in Mathematics 260, Springer, 2011. · Zbl 1206.13001  J. Herzog, T. Hibi and H. Ohsugi, Binomial Ideals, Graduate Texts in Mathematics 279, Springer, 2018. · Zbl 1403.13004  Ohsugi, H.; Hibi, T., Normal polytopes arising from finite graphs, J. Algebra, 207, 409-426 (1998) · Zbl 0926.52017  Stanley, RP, A monotonicity property of $$h$$-vectors and $$h^*$$-vectors, Europ. J. Combin., 14, 251-258 (1993) · Zbl 0799.52008  R.H: Villarreal, Monomial Algebras (Second Edition), Chapman and Hall/CRC, 2015. · Zbl 1325.13004
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