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On Generalisations of the AVD Conjecture to Digraphs

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Abstract

Given an undirected graph, in the AVD (edge-colouring) Conjecture, the goal is to find a proper edge-colouring with the least number of colours such that every two adjacent vertices are incident to different sets of colours. More precisely, the conjecture says that, a few exceptions apart, every graph G should admit such an edge-colouring with at most \(\Delta (G)+2\) colours. Several aspects of interest behind this problem have been investigated over the recent years, including verifications of the conjecture for particular graph classes, general approximations of the conjecture, and multiple generalisations. In this paper, following a recent work of Sopena and Woźniak, generalisations of the AVD Conjecture to digraphs are investigated. More precisely, four of the several possible ways of generalising the conjecture are focused upon. We completely settle one of our four variants, while, for the three remaining ones, we provide partial results.

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Authors and Affiliations

  1. Université Côte d’Azur, CNRS, Inria, I3S, 06902, Sophia-Antipolis, France

    Julien Bensmail

  2. Laboratoire d’Informatique et Systèmes, Aix-Marseille Université, CNRS, Marseille, France

    Fionn Mc Inerney

  3. Université de Toulon Faculté des Sciences de Luminy, Marseille, France

    Fionn Mc Inerney

Authors
  1. Julien Bensmail
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  2. Fionn Mc Inerney
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Correspondence to Julien Bensmail.

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This work has been partially supported by the ANR project DISTANCIA (ANR-14-CE25-0006).

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Bensmail, J., Mc Inerney, F. On Generalisations of the AVD Conjecture to Digraphs. Graphs and Combinatorics 37, 545–558 (2021). https://doi.org/10.1007/s00373-020-02263-9

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  • DOI: https://doi.org/10.1007/s00373-020-02263-9

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