Regularity of symbolic powers of cover ideals of graphs

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 $$\begin{aligned} k\mathrm{deg}(J(G))\le \mathrm{reg}(J(G)^{(k)})\le (k-1)\mathrm{deg}(J(G))+|V(G)|-1. \end{aligned}$$ We also determine families of graphs for which the above inequalities are equality.