On Generalisations of the AVD Conjecture to Digraphs


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.

This is a preview of subscription content, access via your institution.


  1. 1.

    Akbari, S., Bidkhori, H., Nosrati, N.: r-strong edge colorings of graphs. Discr. Math. 306, 3005–3010 (2006)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Balister, P.N., Győri, E., Lehel, J., Schelp, R.H.: Adjacent vertex distinguishing edge-colorings. SIAM J. Discr. Math. 21, 237–250 (2007)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Barme, E., Bensmail, J., Przybyło, J., Woźniak, M.: On a directed variation of the 1-2-3 and 1–2 Conjectures. Discr. Appl. Math. 217, 123–131 (2017)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Baudon, O., Bensmail, J., Sopena, É.: An oriented version of the 1–2–3 conjecture. Discuss. Math. Graph Theory 35(1), 141–156 (2015)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Bensmail, J., Szabo Lyngsie, K.: 1–2–3 Conjecture in digraphs: more results and directions. Discr. Appl. Math. 284, 124–137 (2020)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Borowiecki, M., Grytczuk, J., Pilśniak, M.: Coloring chip configurations on graphs and digraphs. Inf. Process. Lett. 112, 1–4 (2012)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Burris, A.C., Schelp, R.H.: Vertex-distinguishing proper edge-colorings. J. Graph Theory 26(2), 73–82 (1997)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Černy, J., Horňak, M., Soták, R.: Observability of a graph. Math. Slovaca 46, 21–31 (1996)

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Erdős, P., Nešetřil, J.: Irregularities of partitions. G. Halász, V.T. Sós, Eds., [Problem], pp. 162–163 (1989)

  10. 10.

    Holyer, I.J.: The NP-completeness of edge coloring. SIAM J. Comput. 10, 718–720 (1981)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Horňak, M., Przybyło, J., Woźniak, M.: A note on a directed version of the 1–2–3 conjecture. Discr. Appl. Math. 236, 472–476 (2018)

    MathSciNet  Article  Google Scholar 

  12. 12.

    König, D.: Über Graphen und ihre Anwendung auf Determinantentheorie und Mengenlehre. Math. Ann. 77(4), 453–465 (1916)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Li, H., Bai, Y., He, W., Sun, Q.: Vertex-distinguishing proper arc colorings of digraphs. Discr. Appl. Math. 209, 276–286 (2016)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Seamone, B.: The 1–2–3 Conjecture and related problems: a survey. Preprint, 2012. arXiv:1211.5122

  15. 15.

    Sopena, É., Woźniak, M.: A note on the neighbour-distinguishing index of digraphs. Preprint, 2019. Available online at arXiv:1909.10240

  16. 16.

    Vizing, V.G.: On an estimate of the chromatic class of a \(p\)-graph. Diskret. Analiz. 3, 25–30 (1964)

    MathSciNet  Google Scholar 

  17. 17.

    Vizing, V.G.: The chromatic class of a multigraph. Kibernetika 3, 29–39 (1965)

    MathSciNet  Google Scholar 

  18. 18.

    West, D.B.: Introduction to Graph Theory. Prentice Hall, New Jersey (1996)

    Google Scholar 

  19. 19.

    Zhang, Z., Liu, L., Wang, J.: Adjacent strong edge coloring of graphs. Appl. Math. Lett. 15, 623–626 (2002)

    MathSciNet  Article  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Julien Bensmail.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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

Download citation


  • AVD conjecture
  • Proper edge-colourings
  • Digraph