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Upper Bounds on the Average Number of Colors in the Non-equivalent Colorings of a Graph

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Abstract

A coloring of a graph is an assignment of colors to its vertices such that adjacent vertices have different colors. Two colorings are equivalent if they induce the same partition of the vertex set into color classes. Let \({\mathcal {A}}(G)\) be the average number of colors in the non-equivalent colorings of a graph G. We give a general upper bound on \({\mathcal {A}}(G)\) that is valid for all graphs G and a more precise one for graphs G of order n and maximum degree \(\Delta (G)\in \{1,2,n-2\}\).

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Acknowledgements

The authors thank Julien Poulain for his precious help in optimizing our programs allowing us to check our conjectures on a large number of graphs.

Funding

Computational resources have been provided by the Consortium des Équipements de Calcul Intensif (CÉCI), funded by the Fonds de la Recherche Scientifique de Belgique (F.R.S.-FNRS) under Grant No. 2.5020.11 and by the Walloon Region.

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Authors and Affiliations

  1. Department of Mathematics and Industrial Engineering, Polytechnique Montréal-Gerad, Montréal, Canada

    Alain Hertz

  2. Computer Science Department-Algorithms Lab, University of Mons, Mons, Belgium

    Hadrien Mélot, Sébastien Bonte, Gauvain Devillez & Pierre Hauweele

Authors
  1. Alain Hertz
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  2. Hadrien Mélot
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  3. Sébastien Bonte
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  4. Gauvain Devillez
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  5. Pierre Hauweele
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Correspondence to Alain Hertz.

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Hertz, A., Mélot, H., Bonte, S. et al. Upper Bounds on the Average Number of Colors in the Non-equivalent Colorings of a Graph. Graphs and Combinatorics 39, 49 (2023). https://doi.org/10.1007/s00373-023-02637-9

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  • DOI: https://doi.org/10.1007/s00373-023-02637-9

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