Advertisement

Adjacent Vertex Distinguishing Edge Coloring of Planar Graphs Without 4-Cycles

  • Danjun HuangEmail author
  • Xiaoxiu Zhang
  • Weifan Wang
  • Ping Wang
Article
  • 20 Downloads

Abstract

The adjacent vertex distinguishing edge coloring of a graph G is a proper edge coloring of G such that the edge coloring set on any pair of adjacent vertices is distinct. The minimum number of colors required for an adjacent vertex distinguishing edge coloring of G is denoted by \(\chi _{a}'(G)\). It is observed that \(\chi _a'(G)\ge \Delta (G)+1\) when G contains two adjacent vertices of degree \(\Delta (G)\). In this paper, we prove that if G is a planar graph without 4-cycles, then \(\chi _a'(G)\le \max \{9,\Delta (G)+1\}\).

Keywords

Adjacent vertex distinguishing edge coloring Planar graph Cycle 

Mathematics Subject Classification

05C15 

Notes

Acknowledgements

This research was supported by Zhejiang Provincial Natural Science Foundation of China under Grant No. LY18A010014 (Danjun Huang), supported partially by NSFC under Grant No. 11771402 (Weifan Wang).

References

  1. 1.
    Akbari, S., Nosrati, H., Bidkhori, N.: \(r\)-Strong edge colorings of graphs. Discret. Math. 306, 3005–3010 (2006)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Balister, P.N., Györi, E., Lehel, J., Schelp, R.H.: Adjacent vertex distinguishing edge colorings. SIAM J. Discret. Math. 21, 237–250 (2007)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bu, Y., Lih, K.-W., Wang, W.: Adjacent vertex distinguishing edge-colorings of planar graphs with grith at least six. Discuss. Math. Graph Theory 31, 429–439 (2011)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Hatami, H.: \(\Delta +300\) is a bound on the adjacent vertex distinguishing edge chromatic number. J. Comb. Theory Ser. B 95, 246–256 (2005)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Hocquard, H., Montassier, M.: Adjacent vertex-distinguishing edge coloring of graph with maximum degree \(\Delta \). J. Comb. Optim. 26, 152–160 (2013)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Huang, D., Miao, Z., Wang, W.: Adjacent vertex distinguishing indices of planar graphs without 3-cycles. Discret. Math. 338, 139–148 (2015)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Wang, W., Wang, Y.: Adjacent vertex distinguishing edge-coloring of graph with smaller maximum average degree. J. Comb. Optim. 19, 471–485 (2010)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Wang, W., Wang, Y.: Adjacent vertex-distinguishing edge colorings of \(K_4\)-minor free graphs. Appl. Math. Lett. 24, 2034–2037 (2011)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Wang, W., Huang, D.: A characterization on the adjacent vertex distinguishing index of planar graphs with large maximum degree. SIAM J. Discret. Math. 29, 2412–2431 (2015)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Yan, C., Huang, D., Chen, D., Wang, W.: Adjacent vertex distinguishing edge colorings of planar graphs with girth at least five. J. Comb. Optim. 28, 893–909 (2014)MathSciNetCrossRefGoogle Scholar
  11. 11.
    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
  2. 2.Department of Mathematics, Statistics and Computer ScienceSt. Francis Xavier UniversityAntigonishCanada

Personalised recommendations