On the Castelnuovo-Mumford regularity of symbolic powers of cover ideals

TL;DR

This paper establishes sharp bounds and exact formulas for the Castelnuovo-Mumford regularity of symbolic powers of cover ideals in graphs, linking algebraic invariants to graph combinatorics.

Contribution

It provides new bounds and exact regularity computations for symbolic powers of cover ideals based on graph properties, including star packing and specific graph classes.

Findings

Sharp upper bounds for regularity in terms of star packing number

Exact regularity for doubly Cohen-Macaulay graphs

Regularity formulas for Cameron-Walker and certain claw-free graphs

Abstract

Assume that $G$ is a graph with cover ideal $J(G)$. For every integer $k\geq 1$, we denote the $k$-th symbolic power of $J(G)$ by $J(G)^{(k)}$. We provide a sharp upper bound for the regularity of $J(G)^{(k)}$ in terms of the star packing number of $G$. Also, for any integer $k\geq 2$, we study the difference between ${\rm reg}(J(G)^{(k)})$ and ${\rm reg}(J(G)^{(k-2)})$. As a consequence, we compute the regularity of $J(G)^{(k)}$ when $G$ is a doubly Cohen-Macaulay graph. Furthermore, we determine ${\rm reg}(J(G)^{(k)})$ if $G$ is either a Cameron-Walker graph or a claw-free graph which has no cycle of length other that $3$ and $5$.