Certificates of Factorisation for a Class of Triangle-Free Graphs

Abstract

The chromatic polynomial $P(G,\lambda)$ gives the number of $\lambda$-colourings of a graph. If $P(G,\lambda)=P(H_{1},\lambda)P(H_{2},\lambda)/P(K_{r},\lambda)$, then the graph $G$ is said to have a chromatic factorisation with chromatic factors $H_{1}$ and $H_{2}$. It is known that the chromatic polynomial of any clique-separable graph has a chromatic factorisation. In this paper we construct an infinite family of graphs that have chromatic factorisations, but have chromatic polynomials that are not the chromatic polynomial of any clique-separable graph. A certificate of factorisation, that is, a sequence of rewritings based on identities for the chromatic polynomial, is given that explains the chromatic factorisations of graphs from this family. We show that the graphs in this infinite family are the only graphs that have a chromatic factorisation satisfying this certificate and having the odd cycle $C_{2n+1}$, $n\geq2$, as a chromatic factor.