A lower bound for depths of powers of edge ideals

Abstract

Let G be a graph, and let I be the edge ideal of G. Our main results in this article provide lower bounds for the depth of the first three powers of I in terms of the diameter of G. More precisely, we show that depth R/I t ≥ d−4t+5where d is the diameter of G and p is the number of connected components of G and 1 ≤ t ≤ 3. For general powers of edge ideals we show that depth R/I t ≥ p − t. As an application of our results we obtain the corresponding lower bounds for the Stanley depth of the first three powers of I .

Regularity and projective dimension of powers of edge ideal of the disjoint union of some weighted oriented gap-free bipartite graphs

“…In this regard, there has been an interest in determining the smallest value t 0 such that pd (S/I t ) is a constant for all t ≥ t 0 . (see [14,19,21,25]).…”

“…If w j = 1 for all j, then I(D) recovers the usual edge ideal of its (undirected) underlying graph. Edge ideals of (undirected) graphs have been investigated extensively in the literature [1,2,3,4,5,6,7,14,19,21,24,25,29]. In general, edge ideals of weighted oriented graphs are different from edge ideals of edge-weighted (undirected) graphs defined by Paulsen and Sather-Wagstaff [27].…”

Regular Sequences On Squares Of Monomial Ideals

“…In the study of the depth function, it is desirable to know the depth of powers of an ideal instead of just the depth of the ideal itself (cf. [1,5,7,9,10,11,13,16]). This motivates the following natural question: when does a linear form or a sequence of linear forms constructed in [6] remain regular or initially regular with respect to powers of I?…”

Some algebraic invariants of the residue class rings of the edge ideals of perfect semiregular trees

“…Herzog et al showed in [16] that the invariant Stanley depth of J/I is combinatorial in nature, where I ⊂ J ⊂ S are monomial ideals. But interestingly it shares some properties and bounds with homological invariant depth; see for instance [6,12,14,16,24,27]. Recall that, a graded free resolution for a finitely generated graded module M ⊂ S is a free resolution of M of the form:…”