Certificates of Factorisation for Chromatic Polynomials

The chromatic polynomial gives the number of proper $\lambda$-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 $P({G},\lambda)=P({H_{1}},\lambda)P({H_{2}},\lambda)/P({K_{r}},\lambda)$ 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 $n^{2}2^{n^{2}/2}$ for the lengths of these certificates, and find much smaller certificates for all chromatic factorisations of graphs of order $\leq 9$.