Algebraic Properties of Chromatic Roots

A 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.

Algebraic Integers as Chromatic and Domination Roots

Let G be a simple graph of order n and λ∈ℕ. A mapping f:V(G)→{1,2,…,λ} is called a λ-colouring of G if f(u)≠f(v) whenever the vertices u and v are adjacent in G. The number of distinct λ-colourings of G, denoted by P(G,λ), is called the chromatic polynomial of G. The domination polynomial of G is the polynomial D(G,λ)=∑i=1nd(G,i)λi, where d(G,i) is the number of dominating sets of G of size i. Every root of P(G,λ) and D(G,λ) is called the chromatic root and the domination root of G, respectively. Since chromatic polynomial and domination polynomial are monic polynomial with integer coefficients, its zeros are algebraic integers. This naturally raises the question: which algebraic integers can occur as zeros of chromatic and domination polynomials? In this paper, we state some properties of this kind of algebraic integers.

Chromatic roots as algebraic integers

International audience A chromatic root is a zero of the chromatic polynomial of a graph. At a Newton Institute workshop on Combinatorics and Statistical Mechanics in 2008, two conjectures were proposed on the subject of which algebraic integers can be chromatic roots, known as the ``$α +n$ conjecture'' and the ``$nα$ conjecture''. These say, respectively, that given any algebraic integer α there is a natural number $n$ such that $α +n$ is a chromatic root, and that any positive integer multiple of a chromatic root is also a chromatic root. By computing the chromatic polynomials of two large families of graphs, we prove the $α +n$ conjecture for quadratic and cubic integers, and show that the set of chromatic roots satisfying the nα conjecture is dense in the complex plane. Une racine chromatique est un zéro du polynôme chromatique d'un graphe. A un atelier au Newton Institute sur la combinatoire et la mécanique statistique en 2008, deux conjectures ont été proposées dont le sujet des entiers algébriques peut être racines chromatiques, connus sous le nom ``la conjecture $α + n$'' et ``la conjecture $n α$ ''. Les conjectures veulent dire, respectivement, que pour chaque entier algébrique $α$ il y a un nombre entier naturel $n$, tel que $α + n$ est une racine chromatique, et que chaque multiple entier positif d'une racine chromatique est aussi une racine chromatique . En calculant les polynômes chromatiques de deux grandes familles de graphes, on prouve la conjecture $α + n$ pour les entiers quadratiques et cubiques, et montre que l'ensemble des racines chromatiques qui confirme la conjecture $nα$ est dense dans le plan complexe.