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Adjacent vertex distinguishing total choosability of planar graphs with maximum degree at least 10

  • Yulin Chang
  • Qiancheng Ouyang
  • Guanghui WangEmail author
Article
  • 46 Downloads

Abstract

A (proper) total-k-coloring \(\phi :V(G)\cup E(G)\rightarrow \{1,2,\ldots ,k\}\) is called adjacent vertex distinguishing if \(C_{\phi }(u)\ne C_{\phi }(v)\) for each edge \(uv\in E(G)\), where \(C_{\phi }(u)\) is the set of the color of u and the colors of all edges incident with u. We use \(\chi ''_a(G)\) to denote the smallest value k in such a coloring of G. Zhang et al. first introduced this coloring and conjectured that \(\chi ''_a(G)\le \Delta (G)+3\) for any simple graph G. For the list version of this coloring, it is known that \(ch''_a(G)\le \Delta (G)+3\) for any planar graph with \(\Delta (G)\ge 11\), where \(ch''_a(G)\) is the adjacent vertex distinguishing total choosability. In this paper, we show that if G is a planar graph with \(\Delta (G)\ge 10\), then \(ch''_a(G)\le \Delta (G)+3\).

Keywords

Adjacent vertex distinguishing total coloring Planar graph Combinatorial Nullstellensatz Discharging 

Notes

Acknowledgements

This work was supported by the National Natural Science Foundation of China (11471193, 11631014), the Foundation for Distinguished Young Scholars of Shandong Province (JQ201501) and Qilu scholar award of Shandong University.

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of MathematicsShandong UniversityJinanPeople’s Republic of China

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