7 research outputs found
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 graph. We prove that the maximum number of -edge-colourings of a -regular graph on vertices is . Our proof is constructible and leads to a branching algorithm enumerating all the -edge-colourings of a -regular graph using a time and polynomial space. In particular, we obtain a algorithm on time and polynomial space to enumerate all the -edge colourings of a cubic graph, improving the running time of of the algorithm due to Golovach et al.~\cite{GKC10}. We also show that the number of -total-colourings of a connected cubic graph is at most . Again, our proof yields a branching algorithm to enumerate all the -total-colourings of a connected cubic graph