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 · ((k − 1)!). 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)!)) and polynomial space. In particular, we obtain a algorithm to enumerate all the 3-edge-colourings of a connected cubic graph in time O∗(2n/2) = O∗(1.4143n) and polynomial space. This improves the running time of O∗(1.5423n) of the algorithm due to Golovach et al. [10]. We also show that the number of 4-total-colourings of a connected cubic graph is at most 3 · 2. Again, our proof yields a branching algorithm to enumerate all the 4-total-colourings of a connected cubic graph.