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

Abstract

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