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Neighbor sum distinguishing total colorings of IC-planar graphs with maximum degree 13

  • Chao Song
  • Changqing XuEmail author
Article

Abstract

A graph is IC-planar if it admits a drawing on the plane with at most one crossing per edge, such that two pairs of crossing edges share no common end vertex. For a given graph G, a proper total coloring \(\phi \)\(V(G)~\cup ~E(G)\rightarrow \{1,2,\ldots ,k\}\) is called neighbor sum distinguishing if \(f_{\phi }(u)\ne f_{\phi }(v)\) for each \(uv\in E(G)\), where \(f_{\phi }(u)\) is the sum of the color of u and the colors of the edges incident with u. The smallest integer k in such a coloring of G is the neighbor sum distinguishing total chromatic number, denoted by \(\chi ''_{\Sigma }(G)\). Pilśniak and Woźniak conjectured \(\chi _{\Sigma }''(G)\le \Delta (G)+3\) for any simple graph with maximum degree \(\Delta (G)\). This conjecture was confirmed for IC-planar graph with maximum degree at least 14. In this paper, by using the discharging method, we prove that this conjecture holds for any IC-planar graph G with \(\Delta (G)=13\).

Keywords

Neighbor sum distinguishing total coloring IC-planar graph Discharging method 

Notes

Acknowledgements

We would like to thank the referees for their valuable comments. This work was supported by the National Natural Science Foundation of China (11671232).

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

Authors and Affiliations

  1. 1.School of ScienceHebei University of TechnologyTianjinChina

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