Stability of Depths of Powers of Edge Ideals

TL;DR

This paper investigates the stability of the depth function of powers of edge ideals of graphs, providing explicit bounds and conditions under which the depth stabilizes, especially for graphs without 4-cycles.

Contribution

It offers an explicit upper bound for the stabilization point of the depth of powers of edge ideals and characterizes when this bound is tight based on graph structure.

Findings

Derived an explicit upper bound for depth stability point.

Computed the exact limit of depth for certain classes of graphs.

Identified conditions on graphs for the bound to be achieved.

Abstract

Let $G$ be a graph and let $I := I (G)$ be its edge ideal. In this paper, we provide an upper bound of $n$ from which $\depth R/ I(G)^n$ is stationary, and compute this limit explicitly. This bound is always achieved if $G$ has no cycles of length $4$ and every its connected component is either a tree or a unicyclic graph.