Improved bounds for the regularity of edge ideals of graphs

Abstract

Let G be a graph with n vertices, let $S={\mathbb {K}}[x_1,\dots ,x_n]$ be the polynomial ring in n variables over a field ${\mathbb {K}}$ and let I(G) denote the edge ideal of G. For every collection ${\mathcal {H}}$ of connected graphs with $K_2\in {\mathcal {H}}$, we introduce the notions of ${{\mathrm{ind-match}}}_{{\mathcal {H}}}(G)$ and ${{\mathrm{min-match}}}_{{\mathcal {H}}}(G)$. It will be proved that the inequalities ${{\mathrm{ind-match}}}_{\{K_2, C_5\}}(G)\le \mathrm{reg}(S/I(G))\le {{\mathrm{min-match}}}_{\{K_2, C_5\}}(G)$ are true. Moreover, we 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, we prove that if G is a paw-free and doubly Cohen–Macaulay graph, then $\mathrm{reg}(S/I(G))={{\mathrm{ind-match}}}_{\{K_2, C_5\}}(G)$ if and only if every connected component of G is either a complete graph or a 5-cycle graph. Among other results, we show that for every doubly Cohen–Macaulay simplicial complex, the equality $\mathrm{reg}({\mathbb {K}}[\Delta ])=\mathrm{dim}({\mathbb {K}}[\Delta ])$ holds.