On the minimal free resolution of symbolic powers of cover ideals of graphs

TL;DR

This paper characterizes graphs whose symbolic powers of cover ideals have linear resolutions, shows the nondecreasing nature of their regularity sequence, and computes the maximal degree of generators for specific graph classes.

Contribution

It provides a complete characterization of graphs with linear resolution for symbolic powers of cover ideals and computes generator degrees for unmixed and claw-free graphs.

Findings

Graphs with linear resolution of symbolic powers are characterized.

The regularity sequence of symbolic powers is nondecreasing.

Maximal degree of generators is computed for unmixed and claw-free graphs.

Abstract

For any graph $G$, assume that $J(G)$ is the cover ideal of $G$. Let $J(G)^{(k)}$ denote the $k$th symbolic power of $J(G)$. We characterize all graphs $G$ with the property that $J(G)^{(k)}$ has a linear resolution for some (equivalently, for all) integer $k\geq 2$. Moreover, it is shown that for any graph $G$, the sequence $\big({\rm reg}(J(G)^{(k)})\big)_{k=1}^{\infty}$ is nondecreasing. Furthermore, we compute the largest degree of minimal generators of $J(G)^{(k)}$ when $G$ is either an unmixed of a claw-free graph.