Geometric vertex decomposition and liaison for toric ideals of graphs

TL;DR

This paper explores when toric ideals of graphs are geometrically vertex decomposable, establishing new results for bipartite graphs and proposing conjectures for broader classes, with implications for algebraic and combinatorial properties.

Contribution

It introduces conditions under which toric ideals of graphs are geometrically vertex decomposable, including bipartite graphs, and proposes a conjecture linking square-free degenerations to this property.

Findings

Toric ideals of bipartite graphs are geometrically vertex decomposable.

A graph operation preserves geometric vertex decomposability.

Proved the conjecture for graphs with quadratic universal Gr"obner bases.

Abstract

The geometric vertex decomposability property for polynomial ideals is an ideal-theoretic generalization of the vertex decomposability property for simplicial complexes. Indeed, a homogeneous geometrically vertex decomposable ideal is radical and Cohen-Macaulay, and is in the Gorenstein liaison class of a complete intersection (glicci). In this paper, we initiate an investigation into when the toric ideal $I_G$ of a finite simple graph $G$ is geometrically vertex decomposable. We first show how geometric vertex decomposability behaves under tensor products, which allows us to restrict to connected graphs. We then describe a graph operation that preserves geometric vertex decomposability, thus allowing us to build many graphs whose corresponding toric ideals are geometrically vertex decomposable. Using work of Constantinescu and Gorla, we prove that toric ideals of bipartite graphs are geometrically vertex decomposable. We also propose a conjecture that all toric ideals of graphs with a square-free degeneration with respect to a lexicographic order are geometrically vertex decomposable. As evidence, we prove the conjecture in the case that the universal Gr\"obner basis of $I_G$ is a set of quadratic binomials. We also prove that some other families of graphs have the property that $I_G$ is glicci.