Componentwise Linearity of Powers of Cover Ideals

Abstract

Let G be a finite simple graph and J(G) denote its vertex cover ideal in a polynomial ring over a field. Assume that J(G) (k) is its k-th symbolic power. In this paper, we give a criteria for cover ideals of vertex decomposable graphs to have the property that all their symbolic powers are not componentwise linear. Also, we give a necessary and sufficient condition on G so that J(G) (k) is a componentwise linear ideal for some (equivalently, for all) k ≥ 2 when G is a graph such that G \ N G [A] has a simplicial vertex for any independent set A of G. Using this result, we prove that J(G) (k) is a componentwise linear ideal for several classes of graphs for all k ≥ 2. In particular, if G is a bipartite graph, then J(G) is a componentwise linear ideal if and only if J(G) k is a componentwise linear ideal for some (equivalently, for all) k ≥ 2.