Toric ideals associated with gap-free graphs

TL;DR

This paper investigates the algebraic properties of toric ideals associated with gap-free graphs, proving they have squarefree initial ideals and exploring conditions for linear resolutions, with implications for monomial ideals.

Contribution

It establishes that toric ideals of gap-free graphs have squarefree lexicographic initial ideals and characterizes when such ideals have linear resolutions, extending existing results.

Findings

Toric ideals of gap-free graphs have squarefree lexicographic initial ideals.

When the complementary graph is chordal, the toric ideal has a reduced Gr"obner basis with squarefree support.

Monomial ideals generated in degree 2 have linear resolutions if and only if all their powers have linear quotients.

Abstract

In this article we prove that every toric ideal associated with a gap-free graph $G$ has a squarefree lexicographic initial ideal. Moreover, in the particular case when the complementary graph of $G$ is chordal (i.e. when the edge ideal of $G$ has a linear resolution), we show that there exists a reduced Gr\"obner basis $\mathcal{G}$ of the toric ideal of $G$ such that all the monomials in the support of $\mathcal{G}$ are squarefree. Finally, we show (using work by Herzog and Hibi) that if $I$ is a monomial ideal generated in degree 2, then $I$ has a linear resolution if and only if all powers of $I$ have linear quotients, thus extending a result by Herzog, Hibi and Zheng.