On the Stanley depth of powers of edge ideals

TL;DR

This paper investigates the Stanley depth of powers of edge ideals of graphs, establishing lower bounds and conditions under which Stanley's inequality holds, thus advancing understanding of algebraic properties linked to graph structures.

Contribution

It provides new bounds for the Stanley depth of powers of edge ideals and verifies Stanley's inequality for broad classes of these ideals, including specific graph configurations.

Findings

${ m sdepth}(I^k/I^{k+1}) ext{ and } { m sdepth}(S/I^k)$ are at least } p$ for all positive integers $k$.

$S/I^k$ satisfies Stanley's inequality for all $k ext{ with } k ext{ at least } n-1$.

$I^k/I^{k+1}$ satisfies Stanley's inequality for sufficiently large $k$.

Abstract

Let $\mathbb{K}$ be a field and $S=\mathbb{K}[x_1,\dots,x_n]$ be the polynomial ring in $n$ variables over $\mathbb{K}$. 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 ${\rm sdepth}(I^k/I^{k+1})\geq p$ and ${\rm sdepth}(S/I^k)\geq p$ hold. As a consequence, we conclude that $S/I^k$ satisfies the Stanley's inequality for every integer $k\geq n-1$. Also, it follows that $I^k/I^{k+1}$ satisfies the Stanley's inequality for every integer $k\gg 0$. 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 $k\geq n-1$. Moreover, we verify a conjecture of the author in special cases.