Componentwise linearity of ideals arising from graphs

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 $n\ge 3$ 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 $t>1$.

arXiv comment5 pages