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List 2-distance\(\varDelta +3\)-coloring of planar graphs without 4,5-cycles

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Abstract

Let\(\chi _2(G)\) and\(\chi _2^l(G)\) be the 2-distance chromatic number and list 2-distance chromatic number of a graphG, respectively. Wegner conjectured that for each planar graphG with maximum degree\(\varDelta \) at least 4,\(\chi _2(G)\le \varDelta +5\) if\(4\le \varDelta \le 7\), and\(\chi _2(G)\le \lfloor \frac{3\varDelta }{2}\rfloor +1\) if\(\varDelta \ge 8\). LetG be a planar graph without 4,5-cycles. We show that if\(\varDelta \ge 26\), then\(\chi _2^l(G)\le \varDelta +3\). There exist planar graphsG with girth\(g(G)=6\) such that\(\chi _2^l(G)=\varDelta +2\) for arbitrarily large\(\varDelta \). In addition, we also discuss the listL(2, 1)-labeling number ofG, and prove that\(\lambda _l(G)\le \varDelta +8\) for\(\varDelta \ge 27\).

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Acknowledgements

Research supported partially by NSFC (Nos. 61170302, 11601105).

Author information

Authors and Affiliations

  1. Department of Flight Support Command, Air Force Logistics College, Xuzhou, 221000, People’s Republic of China

    Haiyang Zhu, Yu Gu & Jingjun Sheng

  2. Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, People’s Republic of China

    Xinzhong Lü

Authors
  1. Haiyang Zhu

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  2. Yu Gu

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  3. Jingjun Sheng

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  4. Xinzhong Lü

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Correspondence toHaiyang Zhu.

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