Minimum 2-distance coloring of planar graphs and channel assignment

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Abstract

A 2-distance k-coloring of a graph G is a proper k-coloring such that any two vertices at distance two get different colors. \(\chi _{2}(G)\)=min{k|G has a 2-distance k-coloring}. Wegner conjectured that for each planar graph G with maximum degree \(\Delta \), \(\chi _2(G) \le 7\) if \(\Delta \le 3\), \(\chi _2(G) \le \Delta +5\) if \(4\le \Delta \le 7\) and \(\chi _2(G) \le \lfloor \frac{3\Delta }{2}\rfloor +1\) if \(\Delta \ge 8\). In this paper, we prove that: (1) If G is a planar graph with maximum degree \(\Delta \le 5\), then \(\chi _{2}(G)\le 20\); (2) If G is a planar graph with maximum degree \(\Delta \ge 6\), then \(\chi _{2}(G)\le 5\Delta -7\).

Keywords

Planar graph 2-Distance coloring Maximum degree 

Notes

Acknowledgements

The research work was supported by NFSC 11771403.

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.College of Mathematics, Physics and Information EngineeringJiaxing UniversityJiaxingChina
  2. 2.Department of MathematicsZhejiang Normal UniversityJinhuaChina
  3. 3.Zhejiang Normal University Xingzhi CollegeJinhuaChina

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