Depth stability of cover ideals

TL;DR

This paper establishes precise bounds for the stability index of depth functions related to cover ideals of graphs, with exact results for bipartite graphs and forests, advancing understanding of algebraic properties of graph ideals.

Contribution

It provides sharp bounds for the stability index of symbolic and ordinary depth functions of cover ideals, with exact bounds in specific graph classes.

Findings

Sharp bound for symbolic depth stability index sdstab(J)

Exact bound for depth stability index dstab(J) in bipartite graphs

Bound is exact for forests

Abstract

Let R = K[x1,...,xr] be a polynomial ring over a field K. Let G be a graph with vertex set {1,...,r} and let J be the cover ideal of G. We give a sharp bound for the stability index of symbolic depth function sdstab(J). In the case G is bipartite, it yields a sharp bound for the stability index of depth function dstab(J) and this bound is exact if G is a forest.