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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).
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Department of Flight Support Command, Air Force Logistics College, Xuzhou, 221000, People’s Republic of China
Haiyang Zhu, Yu Gu & Jingjun Sheng
Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, People’s Republic of China
Xinzhong Lü
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Zhu, H., Gu, Y., Sheng, J.et al. List 2-distance\(\varDelta +3\)-coloring of planar graphs without 4,5-cycles.J Comb Optim36, 1411–1424 (2018). https://doi.org/10.1007/s10878-018-0312-8
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