Enumerating the edge-colourings and total colourings of a regular graph
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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)!) 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. (Proceedings of WG 2010, pp. 39–50, 2010). We also show that the number of 4-total-colourings of a connected cubic graph is at most 3⋅23n/2. Again, our proof yields a branching algorithm to enumerate all the 4-total-colourings of a connected cubic graph.
KeywordsEdge colouring Total colouring Enumeration (s,t)-ordering Regular graph
The authors would like to thank their children for suggesting them some graph names.
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