Stable value of depth of symbolic powers of edge ideals of graphs

TL;DR

This paper investigates the asymptotic behavior of the depth of symbolic powers of edge ideals in graphs, introducing bipartite connectivity and establishing bounds that are tight for certain graph classes.

Contribution

The paper introduces bipartite connectivity as a new graph invariant and proves an upper bound for the depth of symbolic powers of edge ideals, with exact calculations for specific graph classes.

Findings

Limit of depth of symbolic powers is bounded by bipartite connectivity.

Exact depth values computed for odd cycles and whisker graphs.

Inequality becomes equality for certain classes of graphs.

Abstract

Let $G$ be a simple graph on $n$ vertices. We introduce the notion of bipartite connectivity of $G$, denoted by $\operatorname{bc}(G)$ and prove that $$\lim_{s \to \infty} \operatorname{depth} (S/I(G)^{(s)}) \le \operatorname{bc}(G),$$ where $I(G)$ denotes the edge ideal of $G$ and $S = \mathrm{k}[x_1, \ldots, x_n]$ is a standard graded polynomial ring over a field $\mathrm{k}$. We further compute the depth of symbolic powers of edge ideals of several classes of graphs, including odd cycles and whisker graphs of complete graphs to illustrate the cases where the above inequality becomes equality.