On the Stanley depth of powers of edge ideals

2015/09/16 by Fakhari, S. A. Seyed
#Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.1509.04988

Abstract

Let \mathbbK be a field and S=\mathbbK[x₁,…,x_n] be the polynomial ring in n variables over \mathbbK. 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 \rm sdepth(I^k/I^k+1)≥ p and \rm sdepth(S/I^k)≥ p hold. As a consequence, we conclude that S/I^k satisfies the Stanley's inequality for every integer k≥ n-1. Also, it follows that I^k/I^k+1 satisfies the 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 I^k satisfies the Stanley's inequality for every integer k≥ n-1. Moreover, we verify a conjecture of the author in special cases.

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