The Coloring Ideal and Coloring Complex of a Graph

TL;DR

This paper introduces a monomial ideal associated with a graph's proper colorings, linking algebraic structures to graph colorings and orientations, and explicitly describes the related simplicial complex.

Contribution

It defines a new monomial ideal in the Stanley-Reisner ring that encodes graph colorings and relates its algebraic invariants to combinatorial properties of the graph.

Findings

Hilbert polynomial of the ideal equals the chromatic polynomial

The associated simplicial complex's Euler characteristic equals the number of acyclic orientations

Explicit combinatorial description of the complex C

Abstract

Let $G$ be a simple graph on $d$ vertices. We define a monomial ideal $K$ in the Stanley-Reisner ring $A$ of the order complex of the Boolean algebra on $d$ atoms. The monomials in $K$ are in one-to-one correspondence with the proper colorings of $G$. In particular, the Hilbert polynomial of $K$ equals the chromatic polynomial of $G$. The ideal $K$ is generated by square-free monomials, so $A/K$ is the Stanley-Reisner ring of a simplicial complex $C$. The $h$-vector of $C$ is a certain transformation of the tail $T(n)= n^d-k(n)$ of the chromatic polynomial $k$ of $G$. The combinatorial structure of the complex $C$ is described explicitly and it is shown that the Euler characteristic of $C$ equals the number of acyclic orientations of $G$.