On the index of depth stability of symbolic powers of cover ideals of graphs

TL;DR

This paper establishes a combinatorial upper bound for the depth stability index of symbolic powers of cover ideals of graphs, computes their depths for specific graph classes, and links regularity to induced matching numbers.

Contribution

It introduces a new combinatorial bound for the depth stability index of symbolic powers of cover ideals and relates regularity to induced matchings in graphs.

Findings

Computed depths of symbolic powers for fully clique-whiskered graphs

Established a bound for the depth stability index of cover ideals

Linked Castelnuovo--Mumford regularity to induced matching numbers

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 $I(G)$ and $J(G)$ denote the edge ideal and the cover ideal of $G$, respectively. We provide a combinatorial upper bound for the index of depth stability of symbolic powers of $J(G)$. As a consequence, we compute the depth of symbolic powers of cover ideals of fully clique-whiskered graphs. Meanwhile, we determine a class of graphs $G$ with the property that the Castelnuovo--Mumford regularity of $S/I(G)$ is equal to the induced matching number of $G$.