Resumen de Improved bounds for the regularity of edge ideals of graphs

S. A. Seyed Fakhari, Siamak Yassemi

• 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.