# Componentwise linearity of ideals arising from graphs

V. Crispin Quinonez and E. Emtander

Arxiv ID: 0911.2026•Last updated: 12/7/2021

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 $t>1$.

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