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

Author

Listed:
  • S. Bessy

    (Université Montpellier 2—CNRS, LIRMM)

  • F. Havet

    (Projet Mascotte, I3S (CNRS, UNSA) and INRIA)

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 ∗(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.

Suggested Citation

  • S. Bessy & F. Havet, 2013. "Enumerating the edge-colourings and total colourings of a regular graph," Journal of Combinatorial Optimization, Springer, vol. 25(4), pages 523-535, May.
    • Handle: RePEc:spr:jcomop:v:25:y:2013:i:4:d:10.1007_s10878-011-9448-5
    DOI: 10.1007/s10878-011-9448-5
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    Citations

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    Cited by:

    1. Huijuan Wang & Panos M. Pardalos & Bin Liu, 2019. "Optimal channel assignment with list-edge coloring," Journal of Combinatorial Optimization, Springer, vol. 38(1), pages 197-207, July.
    2. Huijuan Wang & Lidong Wu & Xin Zhang & Weili Wu & Bin Liu, 2016. "A note on the minimum number of choosability of planar graphs," Journal of Combinatorial Optimization, Springer, vol. 31(3), pages 1013-1022, April.
    3. Huijuan Wang & Lidong Wu & Weili Wu & Jianliang Wu, 2014. "Minimum number of disjoint linear forests covering a planar graph," Journal of Combinatorial Optimization, Springer, vol. 28(1), pages 274-287, July.

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