Depth and Stanley depth of symbolic powers of cover ideals of graphs

TL;DR

This paper investigates the depth and Stanley depth of symbolic powers of cover ideals of graphs, proving their non-increasing nature, convergence, and conditions under which Stanley's inequality holds, with implications for algebraic graph theory.

Contribution

It establishes the non-increasing and convergent behavior of depth and Stanley depth sequences for symbolic powers of cover ideals, and proves Stanley's inequality for large powers, providing new insights and an alternative proof for existing results.

Findings

Sequences of Stanley depth are non-increasing and converge.

Stanley's inequality holds for symbolic powers beyond a certain threshold.

Provides an alternative proof for the depth formula of symbolic powers.

Abstract

Let $G$ be a graph with $n$ vertices and let $S=\mathbb{K}[x_1,\dots,x_n]$ be the polynomial ring in $n$ variables over a field $\mathbb{K}$. Assume that $J(G)$ is the cover ideal of $G$ and $J(G)^{(k)}$ is its $k$-th symbolic power. We prove that the sequences $\{{\rm sdepth}(S/J(G)^{(k)})\}_{k=1}^\infty$ and $\{{\rm sdepth}(J(G)^{(k)})\}_{k=1}^\infty$ are non-increasing and hence convergent. Suppose that $\nu_{o}(G)$ denotes the ordered matching number of $G$. We show that for every integer $k\geq 2\nu_{o}(G)-1$, the modules $J(G)^{(k)}$ and $S/J(G)^{(k)}$ satisfy the Stanley's inequality. We also provide an alternative proof for \cite[Theorem 3.4]{hktt} which states that ${\rm depth}(S/J(G)^{(k)})=n-\nu_{o}(G)-1$, for every integer $k\geq 2\nu_{o}(G)-1$.