Symbolic powers of cover ideal of very well-covered and bipartite graphs

TL;DR

This paper investigates the properties of symbolic powers of cover ideals in specific classes of graphs, establishing conditions for linear resolutions, analyzing depth behavior, and providing bounds on regularity.

Contribution

It proves that symbolic powers of cover ideals of very well-covered graphs with linear resolutions also have linear resolutions, and establishes bounds on regularity for bipartite graphs.

Findings

Symbolic powers of cover ideals of certain graphs have linear resolutions.

Depth of symbolic powers decreases or remains constant.

Linear upper bounds for regularity of cover ideals in bipartite graphs.

Abstract

Let $G$ be a graph with $n$ vertices and $S=\mathbb{K}[x_1,\dots,x_n]$ be the polynomial ring in $n$ variables over a field $\mathbb{K}$. Assume that $J(G)$ is the cover ideal of $G$ and $J(G)^{(k)}$ is its $k$-th symbolic power. We prove that if $G$ is a very well-covered graph such that $J(G)$ has linear resolution, then $J(G)^{(k)}$ has linear resolution, for every integer $k\geq 1$. We also prove that for a every very well-covered graph $G$, the depth of symbolic powers of $J(G)$ forms a non-increasing sequence. Finally, we determine a linear upper bound for the regularity of powers of cover ideal of bipartite graph.