Journal of Combinatorial Optimization

, Volume 36, Issue 1, pp 55–64 | Cite as

Minimum 2-distance coloring of planar graphs and channel assignment

  • Junlei Zhu
  • Yuehua Bu


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 \(\chi _{2}(G)\le 5\Delta -7\).


Planar graph 2-Distance coloring Maximum degree 



The research work was supported by NFSC 11771403.


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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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