A note on toric ideals of graphs and Knutson-Miller-Yong decompositions

TL;DR

This paper explores the relationship between graph properties and the algebraic structure of their associated toric ideals using Gr"obner basis techniques, providing new insights into graph bipartiteness and chromatic number bounds.

Contribution

It introduces a novel application of Gr"obner basis methods to analyze toric ideals of graphs, including a height formula and bipartiteness detection.

Findings

Recovered a known height formula for toric ideals of graphs

Identified an algebraic property to detect bipartite graph deletions

Bound the chromatic number using initial ideals

Abstract

We use a Gr\"obner basis technique first introduced by Knutson, Miller and Yong to study the interplay between properties of a graph $G$ and algebraic properties of the toric ideal that it defines. We first recover a well-known height formula for the toric ideal of a graph $I_G$ and demonstrate an algebraic property that can detect when a graph deletion is bipartite. We also bound the chromatic number $\chi(G)$ using information about an initial ideal of $I_G$.