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 ?=?[?1,…,??] be the polynomial ring in n variables over a field ? and let I(G) denote the edge ideal of G. For every collection of connected graphs with ?2∈ , we introduce the notions of ind−match(?) and min−match(?) . It will be proved that the inequalities ind−match{?2,?5}(?)≤reg(?/?(?))≤min−match{?2,?5}(?) are true. Moreover, we show that if G is a Cohen–Macaulay graph with girth at least five, then reg(?/?(?))=ind−match{?2,?5}(?) . Furthermore, we prove that if G is a paw-free and doubly Cohen–Macaulay graph, then reg(?/?(?))=ind−match{?2,?5}(?) 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 reg(?[Δ])=dim(?[Δ]) holds.