Adjacent vertex-distinguishing edge coloring of graphs

  • Marthe Bonamy
  • Nicolas Bousquet
  • Hervé Hocquard
Part of the CRM Series book series (PSNS, volume 16)


An adjacent vertex-distinguishing edge coloring (AVD-coloring) of a graph is a proper edge coloring such that no two neighbors are adjacent to the same set of colors. Zhang et al. [17] conjectured that every connected graph on at least 6 vertices is AVD (Δ + 2)-colorable, where A is the maximum degree. In this paper, we prove that (Δ + 1) colors are enough when A is sufficiently larger than the maximum average degree, denoted mad. We also provide more precise lower bounds for two graph classes: planar graphs, and graphs with mad < 3. In the first case, Δ ≥ 12 suffices, which generalizes the result of Edwards et al. [7] on planar bipartite graphs. No other results are known in the case of planar graphs. In the second case, Δ ≥ 4 is enough, which is optimal and completes the results of Wang and Wang [14] and of Hocquard and Montassier [9].


Planar Graph Discrete Math Graph Class Edge Coloring Chromatic Index 
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  1. [1]
    S. Akbari, H. Bidkhori and N. Nosrati, r-strong edge colorings of graphs, Discrete Math. 306 (2006), 3005–3010.CrossRefzbMATHMathSciNetGoogle Scholar
  2. [2]
    P. N. Balister, E. Győri, J. Lehel and R. H. Schelp, Adjacent vertex distinguishing edge-colorings, SIAM J. Discrete Math. 21 (2007), 237–250.CrossRefzbMATHMathSciNetGoogle Scholar
  3. [3]
    P. N. Balister, O. M. Riordan and R. H. Schelp, Vertex-distinguishing edge-colorings of graphs, J. Graph Theory, 42 (2003), 95–109.CrossRefzbMATHMathSciNetGoogle Scholar
  4. [4]
    O. V. Borodin, A. V. Kostochka and D. R. Woodall, List Edge and List Total Colourings of Multigraphs, J. Comb. Theory, Series B 71(2) (1997), 184–204.CrossRefzbMATHMathSciNetGoogle Scholar
  5. [5]
    A. C. Burris and R. H. Schelp, Vertex-distinguishing proper edge-colorings, J. Graph Theory 26 (1997), 73–83.CrossRefzbMATHMathSciNetGoogle Scholar
  6. [6]
    J. Cerný, M. Horňák and R. Soták, Observability of a graph, Mathematica Slovaca 46(1) (1996), 21–31.zbMATHMathSciNetGoogle Scholar
  7. [7]
    K. Edwards, M. Horňák and M. Woźniak, On the neighbour-distinguishing index of a graph, Graphs and Comb. 22(3) (2006), 341–350.CrossRefzbMATHGoogle Scholar
  8. [8]
    O. Favaron, H. Li and R. H. Schelp, Strong edge coloring of graphs, Discrete Math. 159 (1996), 103–109.CrossRefzbMATHMathSciNetGoogle Scholar
  9. [9]
    H. Hocquard and M. Montassier, Adjacent vertex-distinguishing edge coloring of graphs with maximum degree Δ, DOI: 10.1007/S10878-011-9444-9, 2012.Google Scholar
  10. [10]
    A. V. Kostochka and D. R. Woodall, Choosability conjectures and multicircuits, Discrete Math. 240 (2001), 123–143.CrossRefzbMATHMathSciNetGoogle Scholar
  11. [11]
    D. P. Sanders and Y. Zhao, Planar Graphs of Maximum Degree Seven are Class I, J. Comb. Theory, Series B 83(2) (2001), 201–212.CrossRefzbMATHMathSciNetGoogle Scholar
  12. [12]
    V. G. Vizing, On an estimate of the chromatic class of a p-graph, Metody Diskret. Analiz. 3 (1964), 23–30.Google Scholar
  13. [13]
    V. G. Vizing, Colouring the vertices of a graph with prescribed colours (in russian), Diskret. Analiz. 29 (1976), 3–10.zbMATHMathSciNetGoogle Scholar
  14. [14]
    W. Wang and Y. Wang, Adjacent vertex distinguishing edge-colorings of graphs with smaller maximum average degree, J. Comb. Optim., 19 (2010), 471–485.CrossRefzbMATHMathSciNetGoogle Scholar
  15. [15]
    D. R. Woodall, The average degree of an edge-chromatic critical graph II, J. Graph Theory 56(3) (2007), 194–218.CrossRefzbMATHMathSciNetGoogle Scholar
  16. [16]
    D. R. Woodall, The average degree of a multigraph critical with respect to edge or total choosability, Discrete Math. 310 (2010), 1167–1171.CrossRefzbMATHMathSciNetGoogle Scholar
  17. [17]
    Z. Zhang, L. Liu and J. Wang, Adjacent strong edge coloring of graphs, Appl. Math. Lett. 15 (2002), 623–626.Google Scholar

Copyright information

© Scuola Normale Superiore Pisa 2013

Authors and Affiliations

  • Marthe Bonamy
    • 1
  • Nicolas Bousquet
    • 1
  • Hervé Hocquard
    • 2
  1. 1.LIRMM (Université Montpellier 2)Montpellier CedexFrance
  2. 2.LaBRI (Université Bordeaux 1)Talence CedexFrance

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