Enumerating the edge-colourings and total colourings of a regular graph

In this paper, we are interested in computing the number of edge colourings and total colourings of a connected graph. We prove that the maximum number of $k$-edge-colourings of a connected $k$-regular graph on $n$ vertices is $k\cdot((k-1)!)^{n/2}$. Our proof is constructive and leads to a branching algorithm enumerating all the $k$-edge-colourings of a connected $k$-regular graph in time $O^*(((k-1)!)^{n/2})$ and polynomial space. In particular, we obtain a algorithm to enumerate all the $3$-edge-colourings of a connected cubic graph in time $O^*(2^{n/2})=O^*(1.4143^n)$ and polynomial space. This improves the running time of $O^*(1.5423^n)$ of the algorithm due to Golovach et al.~\cite{GKC10}. We also show that the number of $4$-total-colourings of a connected cubic graph is at most $3\cdot 2^{3n/2}$. Again, our proof yields a branching algorithm to enumerate all the $4$-total-colourings of a connected cubic graph.