On the Stanley depth of powers of edge ideals

Let K be a field and S=K[x1,…,xn] be the polynomial ring in n variables over K. Let G be a graph with n vertices. Assume that I=I(G) is the edge ideal of G and p is the number of its bipartite connected components. We prove that for every positive integer k, the inequalities sdepth(Ik/Ik+1)≥p and sdepth(S/Ik)≥p hold. As a consequence, we conclude that S/Ik satisfies Stanley's inequality for every integer k≥n−1. Also, it follows that Ik/Ik+1 satisfies Stanley's inequality for every integer k≫0. Furthermore, we prove that if (i) G is a non-bipartite graph, or (ii) at least one of the connected components of G is a tree with at least one edge, then Ik satisfies Stanley's inequality for every integer k≥n−1. Moreover, we verify a conjecture of the author in special cases.