Decreasing behavior of the depth functions of edge ideals

TL;DR

This paper investigates the decreasing behavior of depth functions of edge ideals of connected non-bipartite graphs, providing combinatorial conditions for depth values and revealing surprising relationships between depths of different powers.

Contribution

It introduces combinatorial criteria for when the depth of powers of edge ideals equals 1 and demonstrates the rapid decrease of depth to zero, including for symbolic powers.

Findings

Depth function decreases to 0 after reaching 1

If depth of R/I is 1, then depth of R/I^2 is 0

Depth of symbolic powers also exhibits similar behavior

Abstract

Let $I$ be the edge ideal of a connected non-bipartite graph and $R$ the base polynomial ring. Then $\operatorname{depth} R/I \ge 1$ and $\operatorname{depth} R/I^t = 0$ for $t \gg 1$. We give combinatorial conditions for $\operatorname{depth} R/I^t = 1$ for some $t$ in between and show that the depth function is non-increasing thereafter. Especially, the depth function quickly decreases to 0 after reaching 1. We show that if $\operatorname{depth} R/I = 1$ then $\operatorname{depth} R/I^2 = 0$ and if $\operatorname{depth} R/I^2 = 1$ then $\operatorname{depth} R/I^5 = 0$. Other similar results suggest that if $\operatorname{depth} R/I^t = 1$ then $\operatorname{depth} R/I^{t+3} = 0$. This a surprising phenomenon because the depth of a power can determine a smaller depth of another power. Furthermore, we are able to give a simple combinatorial criterion for $\operatorname{depth} R/I^{(t)} = 1$ for $t \gg 1$ and show that the condition $\operatorname{depth} R/I^{(t)} = 1$ is persistent, where $I^{(t)}$ denotes the $t$-th symbolic powers of $I$.