The Scope of Three Colouring Conjectures

It is shown that the Overfull Conjecture, which would provide a chromatic index characterization for a large class of graphs, and the Conformability Conjecture, which would provide a total chromatic number characterization for a large class of graphs, both in fact apply to almost all graphs, whether labelled or unlabelled. The arguments are based on Polya’s theorem, and are elementary in the sense that practically no knowledge of random graph theory is presupposed. It is similarly shown that the Biconformability Conjecture, which would provide a total chromatic number characterization for a large class of equibipartite graphs, in fact applies to almost all equibipartite graphs.