Componentwise linearity of ideals arising from graphs

TL;DR

This paper investigates when intersections of powers of edge ideals from graphs are componentwise linear, showing that for complete graphs they always are, but providing a counterexample for chordal graphs.

Contribution

It extends the understanding of componentwise linearity of intersection ideals from graphs, especially for powers greater than one, with new results for complete graphs and a counterexample for chordal graphs.

Findings

Complete graphs have componentwise linear intersection ideals for all powers.

Counterexample shows chordal graphs may not have componentwise linear ideals for powers > 1.

Provides conditions and examples clarifying when ideals are componentwise linear.

Abstract

Let $G$ be a simple undirected graph on $n$ vertices. Francisco and Van Tuyl have shown that if $G$ is chordal, then $\bigcap_{\{x_i,x_j\}\in E_G} < x_i,x_j>$ is componentwise linear. A natural question that arises is for which $t_{ij}>1$ the ideal $\bigcap_{\{x_i,x_j\}\in E_G}< x_i, x_j>^{t_{ij}}$ is componentwise linear, if $G$ is chordal. In this report we show that $\bigcap_{\{x_i,x_j\}\in E_G} < x_i, x_j>^{t}$ is componentwise linear for all $n\geq 3$ and positive $t$, if $G$ is a complete graph. We give also an example where $G$ is chordal, but the intersection ideal is not componentwise linear for any $t>1$.