Admissible subgraphs and the depth of symbolic powers of cover ideals of graphs

TL;DR

This paper introduces $t$-admissible subgraphs to compute the depth of symbolic powers of cover ideals of graphs, providing explicit formulas for cycles.

Contribution

It defines $t$-admissible subgraphs and applies them to determine the depth of symbolic powers of cover ideals, including a closed-form formula for cycles.

Findings

Derived a formula for the depth of symbolic powers of cover ideals of cycles.

Introduced $t$-admissible subgraphs as a tool for depth computation.

Established a method applicable to various graph classes.

Abstract

Let $G$ be a simple graph. We introduce the notion of $t$-admissible subgraphs of $G$ and show how to use them to compute the depth of the $t$-th symbolic powers of the cover ideal of $G$. As an application, we prove that \[ \depth\big(S/J(C_n)^{(t)}\big) = n - 1 - \left\lfloor \frac{tn}{2t+1} \right\rfloor \] for all $t \ge 2$ and $n \ge 3$, where $S = K[x_1,\ldots,x_n]$ and $J(C_n)$ is the cover ideal of the cycle on $n$ vertices.