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. % $\mathbb{K}$. The $k$-th symbolic power of $J(G)$ is denoted by $J(G{)}^{(k)}$. 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) $k\ge 2$ 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) $k\ge 2$.

arXiv admin note: text overlap with arXiv:1908.10576 To be published in Journal of Algebraic Combinatorics

References in corpus (1)

• Herzog, Hibi and Ohsugi conjecture for trees