Lower bounds for the depth of symbolic powers of edge ideals of graphs

Abstract

Assume that G is a graph with edge ideal I(G). For every integer s≥1, we denote the s-th symbolic power of I(G) by I(G)(s). It is shown that for every very well-covered graph G (without isolated vertices) and for each integer s≥1, we have depthS/I(G)(s)≥2c(G)−p(G), where c(G) and p(G) denote the number of connected components and the number of bipartite connected components of G, respectively. Moreover, for any graph G, we determine a sharp lower bound for the depth of I(G)(3) in terms of the star packing number of G.