Componentwise Linearity of Powers of Cover Ideals

TL;DR

This paper characterizes when powers of vertex cover ideals of certain graphs are componentwise linear, providing criteria for classes like bipartite graphs and establishing conditions for all powers to have this property.

Contribution

It offers new criteria for the componentwise linearity of symbolic and ordinary powers of cover ideals in graphs, especially for vertex decomposable and bipartite graphs.

Findings

Characterizes when all symbolic powers are not componentwise linear.

Provides necessary and sufficient conditions for componentwise linearity of powers.

Shows that for bipartite graphs, the linearity of the ideal's powers is equivalent across all powers.

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) $k \geq 2$ 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 $k \geq 2$. 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) $k \geq 2$.