Algebraic properties of chromatic roots

TL;DR

This paper explores the algebraic properties of chromatic roots, conjecturing their relation to algebraic integers, and reports on theoretical and computational advances in understanding their algebraic nature and Galois groups.

Contribution

It introduces a conjecture linking algebraic integers to chromatic roots, proves it for quadratic integers, and discusses computational results on Galois groups of chromatic polynomial factors.

Findings

Conjecture that every algebraic integer plus a natural number is a chromatic root.

Proved the conjecture for quadratic integers.

Reported computational results on Galois groups of chromatic polynomial factors.

Abstract

A \emph{chromatic root} is a root of the chromatic polynomial of a graph. Any chromatic root is an algebraic integer. Much is known about the location of chromatic roots in the real and complex numbers, but rather less about their properties as algebraic numbers. This question was the subject of a seminar at the Isaac Newton Institute in late 2008. The purpose of this paper is to report on the seminar and subsequent developments. We conjecture that, for every algebraic integer $\alpha$, there is a natural number $n$ such that $\alpha+n$ is a chromatic root. This is proved for quadratic integers, an extension to cubic integers has been found by Adam Bohn. The idea is to consider certain special classes of graphs for which the chromatic polynomial is a product of linear factors and one "interesting" factor of larger degree. We also report computational results on the Galois groups of irreducible factors of the chromatic polynomial for some special graphs. Finally, extensions to the Tutte polynomial are mentioned briefly.