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Reports (Research Report) Year : 2011

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

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.
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Dates and versions

inria-00602188 , version 1 (21-06-2011)

Identifiers

  • HAL Id : inria-00602188 , version 1

Cite

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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