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