Abstract
Let \(\chi _2(G)\) and \(\chi _2^l(G)\) be the 2-distance chromatic number and list 2-distance chromatic number of a graph G, respectively. Wegner conjectured that for each planar graph G 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 \(\varDelta \ge 8\). Let G 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 graphs G with girth \(g(G)=6\) such that \(\chi _2^l(G)=\varDelta +2\) for arbitrarily large \(\varDelta \). In addition, we also discuss the list L(2, 1)-labeling number of G, and prove that \(\lambda _l(G)\le \varDelta +8\) for \(\varDelta \ge 27\).
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Research supported partially by NSFC (Nos. 61170302, 11601105).
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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 Optim 36, 1411–1424 (2018). https://doi.org/10.1007/s10878-018-0312-8
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DOI: https://doi.org/10.1007/s10878-018-0312-8