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

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