Regularity of symbolic powers of cover ideals of graphs

TL;DR

This paper investigates the regularity of symbolic powers of cover ideals in specific graph classes, establishing bounds and characterizing cases of equality, advancing understanding of algebraic properties linked to graph structures.

Contribution

It provides new bounds on the regularity of symbolic powers of cover ideals for bipartite, unmixed, and claw-free graphs, and characterizes when these bounds are tight.

Findings

Established bounds: $k\deg(J(G)) \leq \operatorname{reg}(J(G)^{(k)}) \leq (k-1)\deg(J(G)) + |V(G)| - 1$.

Identified graph families where bounds are achieved with equality.

Enhanced understanding of the algebraic invariants of graph-related ideals.

Abstract

Let $G$ be a graph which belongs to either of the following classes: (i) bipartite graphs, (ii) unmixed graphs, or (iii) claw--free graphs. Assume that $J(G)$ is the cover ideal $G$ and $J(G)^{(k)}$ is its $k$-th symbolic power. We prove that$$k{\rm deg}(J(G))\leq {\rm reg}(J(G)^{(k)})\leq (k-1){\rm deg}(J(G))+|V(G)|-1.$$We also determine families of graphs for which the above inequalities are equality.