Castelnuovo – Mumford regularity of edge ideal of graphs

Let G be a graph with n vertices, S=mathbb{K}[x_1,dots,x_n] be the polynomial ring in n variables over a field mathbb{K} and I(G) denote the edge ideal of G.

We survey a number of recent studies of the Castelnuovo-Mumford regularity of the edge ideal of G. Our focus is on bounds and exact values for the regularity in terms of combinatorial data from associated graphs.

In addition, for every collection mathcal{H} of connected graphs with K_2in mathcal{H}, we introduce the notions of ind-match_{mathcal{H}}(G) and min-match_{mathcal{H}}(G). We will improve the inequalities for regularity of S/I(G).