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. 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) \(k \ge 2\) when G is a graph such that \(G {\setminus } N_G[A]\) 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 \(k \ge 2\). 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) \(k \ge 2\).