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

TL;DR

This work explores the interplay between graph combinatorics and toric ideals via Knutson-Miller-Yong (KMY) decompositions. It provides a new inductive proof of the height formula for $I_G$ and derives a bound on $χ(G)$ from Gröbner degenerations, linking algebraic initial ideals to coloring. The approach uses nondegenerate KMY decompositions along nonbridge edges to reduce to subgraphs and preserve the toric-graph structure, enabling iterative height calculations. The results show how initial ideals and edge deletions encode bipartiteness and coloring information, offering a computational bridge between combinatorics and algebraic geometry.

Abstract

We use a Gröbner 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 $χ(G)$ using information about an initial ideal of $I_G$.

Figures (5)

• Figure 1: A non-bipartite graph $G$ such that $G\setminus y$ is bipartite.
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Theorems & Definitions (40)

• Theorem 1
• Theorem 2
• Theorem 3
• Definition 1.1
• Remark 1.2
• Definition 1.3
• Proposition 1.4
• Remark 1.5
• Theorem 1.6
• proof