Improved bounds for the regularity of edge ideals of graphs (Q1752915)

Let \(S=\mathbb{K}[x_1,\dots,x_n]\) be the polynomial ring in \(n\) variables over a field \(\mathbb{K}\). Let \(G\) be a simple graph with vertex set \(V(G) = \{v_1,\dots,v_n\}\), edge set \(E(G)\) and the edge ideal \(I(G)\). A subset \(M\subseteq E(G)\) is a matching if \(e\cap e'=\emptyset\), for every pair of edges \(e,e'\in M\). The cardinality of the largest matching of \(G\) is called the matching number of \(G\) and is denoted by \(\mathrm{match}(G)\). The minimum cardinality of the maximal matchings of \(G\) is the minimum matching number of \(G\) and is denoted by \(\mathrm{min-match}(G)\). A matching \(M\) of \(G\) is an induced matching of \(G\) if for every pair of edges \(e,e'\in M\) there is no edge \(f\in E(G)\setminus M\) with \(f\subset e\cup e'\). The cardinality of the largest induced matching of \(G\) is called the induced matching number of \(G\) and is denoted by \(\mathrm{ind-match}(G)\). Let \(\mathcal{H}\) be a collection of connected graphs with \(K_2\in\mathcal{H}\). In this article under review, the authors introduce the notions of \(\mathrm{ind-match}_{\mathcal{H}}(G)\) and \(\mathrm{min-match}_{\mathcal H}(G)\) and they prove that the inequalities \(\mathrm{ind-match}_{\{K_2,C_5\}}(G)\leq \mathrm{reg}(S/I(G))\leq\mathrm{min-match}_{\{K_2,C_5\}}(G)\) are true. Moreover, the authors show that if \(G\) is a Cohen-Macaulay graph with girth at least five, then \(\mathrm{reg}(S/I(G))=\mathrm{ind-match}_{\{K_2,C_5\}}(G)\). Furthermore, they show that for every doubly Cohen-Macaulay simplicial complex, the equality \(\mathrm{reg}(\mathbb{K}[\Delta])=\dim(\mathbb{K}[\Delta])\) holds.