Monomial invariants applied to graph coloring

TL;DR

This paper links monomial ideals to graph coloring by showing the chromatic number equals the codimension of a constructed ideal and provides a formula for the chromatic polynomial for certain graphs, including those satisfying the Erdős-Faber-Lovász conjecture.

Contribution

It introduces a novel approach connecting monomial ideals with graph coloring, providing formulas for chromatic polynomials and relating multiplicity and codimension to graph invariants.

Findings

Chromatic number equals the codimension of the associated monomial ideal.

A formula for the chromatic polynomial evaluated at the chromatic number.

Applicability to graphs satisfying the Erdős-Faber-Lovász conjecture.

Abstract

This article is built upon three main ideas. First, for a class of monomial ideals, it is proven that the multiplicity of an ideal equals the number of realizations of its codimension (an intuitive concept that we define later). Next, for an arbitrary graph G, we construct a monomial ideal M_G, and show that the chromatic number of G is equal to the codimension of M_G. Finally, for a class of graphs, we give a formula that computes the chromatic polynomial of G, evaluated at the chromatic number of G, in terms of the codimension and multiplicity of M_G. In particular, the formula applies to all graphs satisfying the Erdos-Faber-Lov\'asz conjecture.