Two-Distance Vertex-Distinguishing Index of Sparse Subcubic Graphs

  • Loumngam Kamga Victor
  • Juan Liu
  • Weifan WangEmail author


The 2-distance vertex-distinguishing index \(\chi '_\mathrm{d2}(G)\) of a graph G is the minimum number of colors required for a proper edge coloring of G such that any pair of vertices at distance two have distinct sets of colors. It was conjectured that every subcubic graph G has \(\chi '_{\mathrm{d2}}(G)\le 5\). In this paper, we confirm this conjecture for subcubic graphs with maximum average degree less than \(\frac{8}{3}\).


Subcubic graph Maximum average degree Edge coloring 2-Distance vertex-distinguishing index AVD edge coloring 

Mathematics Subject Classification




  1. 1.
    Akbari, S., Bidkhori, H., Nosrati, N.: \(r\)-Strong edge colorings of graphs. Discrete Math. 306, 3005–3010 (2006)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Balister, P.N., Győri, E., Lehel, J., Schelp, R.H.: Adjacent vertex distinguishing edge-colorings. SIAM J. Discrete Math. 21, 237–250 (2007)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Hatami, H.: \(\Delta \)+300 is a bound on the adjacent vertex distinguishing edge chromatic number. J. Combin. Theory Ser. B 95, 246–256 (2005)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Huang, D., Lih, K.-W., Wang, W.: Legally \((\Delta +2)\)-coloring bipartite outerplanar graphs in cubic time. In: Lu, Z., Kim, D., Wu, W., Li, W., Du, D.Z. (eds.) Combinatorial Optimization and Applications. Lecture Notes in Comput Sci, vol. 9486, pp. 617–632. Springer, Cham (2015)CrossRefGoogle Scholar
  5. 5.
    Victor, L.K., Wang, W., Wang, Y., Chen, M.: 2-Distance vertex-distinguishing index of subcubic graphs. J. Combin. Optim. 36, 108–120 (2018)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Vučković, B.: Edge-partitions of graphs and their neighbor-distinguishing index. Discrete Math. 340, 3092–3096 (2017)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Wang, W., Huang, D., Wang, Y., Wang, Y., Du, D.Z.: A polynomial-time nearly-optimal algorithm for an edge coloring problem in outerplanar graphs. J. Global Optim. 65, 351–367 (2016)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Wang, W., Wang, Y., Huang, D., Wang Y.: 2-Distance vertex-distinguishing edge coloring of graphs. Preprint (2016)Google Scholar
  9. 9.
    Wang, Y., Wang, W., Huo, J.: Some bounds on the neighbor-distinguishing index of graphs. Discrete Math. 338, 2006–2013 (2015)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Wang, W., Wang, Y.: Adjacent vertex distinguishing edge-colorings of graphs with smaller maximum average degree. J. Combin. Optim. 19, 471–485 (2010)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Zhang, L., Wang, W., Lih, K.-W.: An improved upper bound on the adjacent vertex distinguishing chromatic index of a graph. Discrete Appl. Math. 162, 348–354 (2014)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Zhang, Z., Liu, L., Wang, J.: Adjacent strong edge coloring of graphs. Appl. Math. Lett. 15, 623–626 (2002)MathSciNetCrossRefGoogle Scholar

Copyright information

© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2019

Authors and Affiliations

  1. 1.Department of MathematicsZhejiang Normal UniversityJinhuaChina

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