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(H 1 , λ)P(H 2 , λ)/P(K r , λ) for some graphs H 1 and H 2 and clique K r . 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 n 2 2 n2/2 for the lengths of these certificates, and find much smaller certificates for all chromatic factorisations of graphs of order ≤ 9.