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

Nguyen Cong Minh, Tran Nam Trung and Thanh Vu

Vol. 329 (2024), No. 1, 147–164

DOI: 10.2140/pjm.2024.329.147

Abstract

Let $G$ be a simple graph on $n$ vertices. We introduce the notion of bipartite connectivity of $G$, denoted by $\mathrm{bc}<!--FUNCTION\; APPLICATION-->(G)$ and prove that

$$\underset{s\to \infty }{\lim <!--FUNCTION\; APPLICATION-->}\mathrm{depth}<!--FUNCTION\; APPLICATION-->(S∕I{(G)}^{(s)})\le \mathrm{bc}<!--FUNCTION\; APPLICATION-->(G),$$

where $I(G)$ denotes the edge ideal of $G$ and $S=k<!--FUNCTION\; APPLICATION-->[x_1,\dots <!--FUNCTION\; APPLICATION-->,x_n]$ is a standard graded polynomial ring over a field $k<!--FUNCTION\; APPLICATION-->$. 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.

Dedicated to Professor Ngo Viet Trung on the occasion of his 70th birthday

Keywords

depth of symbolic powers, cycles, stable value of depth

Mathematical Subject Classification

Primary: 13D02

Secondary: 05E40, 13F55

Milestones

Received: 4 September 2023

Revised: 14 April 2024

Accepted: 22 April 2024

Published: 12 June 2024