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 aboutthe location of chromatic roots in the real and complex numbers, butrather less about their properties as algebraic numbers. This questionwas 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 subsequentdevelopments.We conjecture that, for every algebraic integer alpha, there is anatural number n such that alpha+n is a chromatic root. This is provedfor quadratic integers; an extension to cubic integers has been found byAdam Bohn. The idea is to consider certain special classes of graphsfor which the chromatic polynomial is a productof linear factors and one "interesting" factor of larger degree. We alsoreport computational results on the Galois groups of irreducible factorsof the chromatic polynomial for some special graphs. Finally,extensions to the Tutte polynomial are mentioned briefly.