Toric ideals associated with gap-free graphs

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 Gröbner 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.

13 pages, v2. To appear in Journal of Pure and Applied Algebra

References in corpus (2)

• Existence of unimodular triangulations - positive results
• Linear Quotients of the Square of the Edge Ideal of the Anticycle

Cited by in corpus (6)

• Betti numbers of toric ideals of graphs: A case study
• Three invariants of geometrically vertex decomposable ideals
• Lexicographic and reverse lexicographic quadratic Gröbner bases of cut ideals
• Quadratic Generated Normal Domains From Graphs
• Serre's Properties for Quadratic Generated Domains from Graphs
• Existence of a regular unimodular triangulation of the edge polytopes of finite graphs