2-Distance vertex-distinguishing index of subcubic graphs

  • Victor Loumngam Kamga
  • Weifan Wang
  • Ying Wang
  • Min Chen
Article
  • 27 Downloads

Abstract

A 2-distance vertex-distinguishing edge coloring of a graph G is a proper edge coloring of G such that any pair of vertices at distance 2 have distinct sets of colors. The 2-distance vertex-distinguishing index \(\chi ^{\prime }_{\mathrm{d2}}(G)\) of G is the minimum number of colors needed for a 2-distance vertex-distinguishing edge coloring of G. Some network problems can be converted to the 2-distance vertex-distinguishing edge coloring of graphs. It is proved in this paper that if G is a subcubic graph, then \(\chi ^{\prime }_{\mathrm{d2}}(G)\le 6\). Since the Peterson graph P satisfies \(\chi ^{\prime }_{\mathrm{d2}}(P)=5\), our solution is within one color from optimal.

Keywords

Subcubic graph Edge coloring 2-Distance vertex-distinguishing index Star-chromatic index 

References

  1. Aigner M, Triesch E (1990) Irregular assignments of trees and forests. SIAM J Discrete Math 3:439–449MathSciNetCrossRefMATHGoogle Scholar
  2. Akbari S, Bidkhori H, Nosrati N (2006) \(r\)-Strong edge colorings of graphs. Discrete Math 306:3005–3010MathSciNetCrossRefMATHGoogle Scholar
  3. Balister PN, Győri E, Lehel J, Schelp RH (2007) Adjacent vertex distinguishing edge-colorings. SIAM J Discrete Math 21:237–250MathSciNetCrossRefMATHGoogle Scholar
  4. Balister PN, Bollobás B, Schelpa RH (2002) Vertex distinguishing colorings of graphs with \(\Delta (G)=2\). Discrete Math 252:17–29MathSciNetCrossRefMATHGoogle Scholar
  5. Bazgan C, Harkat-Benhamdine A, Li H, Woźniak M (1999) On the vertex-distinguishing proper edge-colorings of graphs. J Comb Theoey Ser B 75:288–301MathSciNetCrossRefMATHGoogle Scholar
  6. Bezegová L, Lužar B, Mockovčiaková M, Soták R, Škrekovski R (2016) Star edge coloring of some classes of graphs. J Graph Theory 81:73–82MathSciNetCrossRefMATHGoogle Scholar
  7. Brooks LR (1941) On colouring the nodes of a network. Proc Camb Philos Soc 37:194–197MathSciNetCrossRefMATHGoogle Scholar
  8. Burris AC (1993) Vertex-distinguishing edge-colorings. Ph.D. Dissertation, Memphis State UniversityGoogle Scholar
  9. Burris AC, Schelp RH (1997) Vertex-distinguishing proper edge-colorings. J Graph Theory 26:73–83MathSciNetCrossRefMATHGoogle Scholar
  10. Dvořák Z, Mohar B, Šámal R (2013) Star chromatic index. J Graph Theory 72:313–326MathSciNetCrossRefMATHGoogle Scholar
  11. Hatami H (2005) \(\Delta \)+300 is a bound on the the adjacent vertex distinguishing edge chromatic number. J Comb Theory Ser B 95:246–256MathSciNetCrossRefMATHGoogle Scholar
  12. Huang D, Lih KW, Wang W (2015) Legally \((\Delta +2)\)-coloring bipartite outerplanar graphs in cubic time. Comb Optim Appl, 617–632. Lecture Notes in Comput Sci 9486, Springer, Cham, 2015Google Scholar
  13. Petersen J (1891) Die Theorie ser regulären Graphsn. Acta Math 15:193–220MathSciNetCrossRefGoogle Scholar
  14. Vučković B (2017) Edge-partitions of graphs and their neighbor-distinguishing index. Discrete Math 340:3092–3096MathSciNetCrossRefMATHGoogle Scholar
  15. Wang W, Huang D, Wang Y, Wang Y, Du DZ (2016) A polynomial-time nearly-optimal algorithm for an edge coloring problem in outerplanar graphs. J Glob Optim 65:351–367MathSciNetCrossRefMATHGoogle Scholar
  16. Wang W, Wang Y, Huang D, Wang Y (2016) 2-Distance vertex-distinguishing edge coloring of graphs (submitted)Google Scholar
  17. Wang Y, Wang W, Huo J (2015) Some bounds on the neighbor-distinguishing index of graphs. Discrete Math 338:2006–2013MathSciNetCrossRefMATHGoogle Scholar
  18. Zhang Z, Li J, Chen X, Cheng H, Yao B (2006) \(D(\beta )\)-vertex-distinguishing proper edge-coloring of graphs. Acta Math Sin (Chin Ser) 49:703–708MathSciNetMATHGoogle Scholar
  19. Zhang Z, Liu L, Wang J (2002) Adjacent strong edge coloring of graphs. Appl Math Lett 15:623–626MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Victor Loumngam Kamga
    • 1
  • Weifan Wang
    • 1
  • Ying Wang
    • 1
  • Min Chen
    • 1
  1. 1.Department of MathematicsZhejiang Normal UniversityJinhuaChina

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