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Journal of Combinatorial Optimization

, Volume 33, Issue 2, pp 779–790 | Cite as

The adjacent vertex distinguishing total coloring of planar graphs without adjacent 4-cycles

  • Lin Sun
  • Xiaohan Cheng
  • Jianliang Wu
Article

Abstract

A total [k]-coloring of a graph G is a mapping \(\phi \): \(V(G)\cup E(G)\rightarrow [k]=\{1, 2,\ldots , k\}\) such that no two adjacent or incident elements in \(V(G)\cup E(G)\) receive the same color. In a total [k]-coloring \(\phi \) of G, let \(C_{\phi }(v)\) denote the set of colors of the edges incident to v and the color of v. If for each edge uv, \(C_{\phi }(u)\ne C_{\phi }(v)\), we call such a total [k]-coloring an adjacent vertex distinguishing total coloring of G. \(\chi ''_{a}(G)\) denotes the smallest value k in such a coloring of G. In this paper, by using the Combinatorial Nullstellensatz and the discharging method, we prove that if a planar graph G with maximum degree \(\Delta \ge 8\) contains no adjacent 4-cycles, then \(\chi ''_{a}(G)\le \Delta +3\).

Keywords

Planar graph Adjacent vertex distinguishing total coloring Combinatorial Nullstellensatz Discharging method 

Mathematics Subject Classification

05C15 

Notes

Acknowledgments

This work is partially supported by NSFC (11271006).

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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of MathematicsChangji UniversityChangjiChina
  2. 2.School of MathematicsShandong UniversityJinanChina

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