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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)].
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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