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

1 ALGCO - Algorithmes, Graphes et Combinatoire
LIRMM - Laboratoire d'Informatique de Robotique et de Microélectronique de Montpellier
2 MASCOTTE - Algorithms, simulation, combinatorics and optimization for telecommunications
CRISAM - Inria Sophia Antipolis - Méditerranée , Laboratoire I3S - COMRED - COMmunications, Réseaux, systèmes Embarqués et Distribués
Abstract : 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 $k$-edge-colourings of a $k$-regular graph on $n$ vertices is $k\cdot(k-1!)^{n/2}$. Our proof is constructible and leads to a branching algorithm enumerating all the $k$-edge-colourings of a $k$-regular graph using a time $O^*((k-1!)^{n/2})$ and polynomial space. In particular, we obtain a algorithm on time $O^*(2^{n/2})=O^*(1.4143^n)$ and polynomial space to enumerate all the $3$-edge colourings of a cubic graph, improving 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 $3.2^{3n/2}$. Again, our proof yields a branching algorithm to enumerate all the $4$-total-colourings of a connected cubic graph.
Document type :
Reports

Cited literature [16 references]

https://hal.inria.fr/inria-00602188
Contributor : Frederic Havet <>
Submitted on : Tuesday, June 21, 2011 - 5:00:40 PM
Last modification on : Monday, October 12, 2020 - 10:30:12 AM
Long-term archiving on: : Friday, November 9, 2012 - 4:46:24 PM

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RR-7652.pdf
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• HAL Id : inria-00602188, version 1

### Citation

Stéphane Bessy, Frédéric Havet. Enumerating the edge-colourings and total colourings of a regular graph. [Research Report] RR-7652, INRIA. 2011. ⟨inria-00602188⟩

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