Journal of Combinatorial Optimization

, Volume 25, Issue 4, pp 523–535

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

Article

## Abstract

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(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. (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.

### Keywords

Edge colouring Total colouring Enumeration (s,t)-ordering Regular graph

## Notes

### Acknowledgement

The authors would like to thank their children for suggesting them some graph names.

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