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Every planar graph with cycles of length neither 4 nor 5 is \((1,1,0)\)-colorable

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Abstract

Let \(d_1, d_2,\dots ,d_k\) be \(k\) non-negative integers. A graph \(G\) is \((d_1,d_2,\ldots ,d_k)\)-colorable, if the vertex set of \(G\) can be partitioned into subsets \(V_1,V_2,\ldots ,V_k\) such that the subgraph \(G[V_i]\) induced by \(V_i\) has maximum degree at most \(d_i\) for \(i=1,2,\ldots ,k.\) Let \(\digamma \) be the family of planar graphs with cycles of length neither 4 nor 5. Steinberg conjectured that every graph of \(\digamma \) is \((0,0,0)\)-colorable. In this paper, we prove that every graph of \(\digamma \) is \((1,1,0)\)-colorable.

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Acknowledgments

The authors are grateful to the referees for their valuable suggestions to improve the presentation of this article. L. Xu and Y. Wang were supported by NSFC No. 11271335 and Z. Miao was supported by NSFC No. 11171288.

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Authors and Affiliations

  1. Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, China

    Lingji Xu & Yingqian Wang

  2. School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, 221116, China

    Zhengke Miao

Authors
  1. Lingji Xu
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  2. Zhengke Miao
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  3. Yingqian Wang
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Correspondence to Yingqian Wang.

Additional information

After this article accepted, the authors were informed that the same result of this paper was independently obtained by Hill and Yu [on arXiv (http://arxiv.org/abs/1208.3395)].

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Xu, L., Miao, Z. & Wang, Y. Every planar graph with cycles of length neither 4 nor 5 is \((1,1,0)\)-colorable. J Comb Optim 28, 774–786 (2014). https://doi.org/10.1007/s10878-012-9586-4

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  • DOI: https://doi.org/10.1007/s10878-012-9586-4

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