Certificates of Factorisation for Chromatic Polynomials

The chromatic polynomial gives the number of proper λ-colourings of a graph G. This paper considers factorisation of the chromatic polynomial as a first step in an algebraic study of the roots of this polynomial. The chromatic polynomial of a graph is said to have a chromatic factorisation if P (G,λ) = P (H1, λ)P (H2, λ)/P (Kr , λ) for some graphs H1 and H2 and clique Kr. It is known that the chromatic polynomial of any clique-separable graph, that is, a graph containing a separating r-clique, has a chromatic factorisation. We show that there exist other chromatic polynomials that have chromatic factorisations but are not the chromatic polynomial of any clique-separable graph and identify all such chromatic polynomials of degree at most 10. We introduce the notion of a certificate of factorisation, that is, a sequence of algebraic transformations based on identities for the chromatic polynomial that explains the factorisations for a graph. We find an upper bound of n22n /2 for the lengths of these certificates, and find much smaller certificates for all chromatic factorisations of graphs of order ≤ 9.