Componentwise linearity of powers of cover ideals

Let G be a finite simple graph and J ( G ) denote its vertex cover ideal in a polynomial ring over a field. The k -th symbolic power of J ( G ) is denoted by J ( G ) ( k ) . In this paper, we give a criterion 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 .