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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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