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

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 \(\chi _a'(G)\le \max \{9,\Delta (G)+1\}\).

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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).

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Correspondence to Danjun Huang.

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Communicated by Xueliang Li.

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Huang, D., Zhang, X., Wang, W. et al. Adjacent Vertex Distinguishing Edge Coloring of Planar Graphs Without 4-Cycles. Bull. Malays. Math. Sci. Soc. 43, 3159–3181 (2020). https://doi.org/10.1007/s40840-019-00860-3

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Keywords

  • Adjacent vertex distinguishing edge coloring
  • Planar graph
  • Cycle

Mathematics Subject Classification

  • 05C15