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List 2-distance coloring of planar graphs

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Abstract

The \(2\)-distance coloring of a graph \(G\) is to color the vertices of \(G\) so that every two vertices at distance at most \(2\) from each other get different colors. Let \(\chi _{2}^{l}(G)\) be the list 2-distance chromatic number of \(G\). In this paper, we show that (1) a planar graph \(G\) with \(\Delta (G)\ge 12\) which contains no \(3,5\)-cycles and intersecting 4-cycles has \(\chi _{2}^{l}(G)\le \Delta +6\); (2) a planar graph \(G\) with \(\Delta (G)\le 5\) and \(g(G)\ge 5\) has \(\chi _{2}^{l}(G)\le 13\).

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References

  • Agnarsson G, Halldorsson MM (2000) Coloring powers of planar graphs. SIAM J Discret Math 16:651–662

    Article  MathSciNet  Google Scholar 

  • Agnarsson G, Halldorsson MM (2003) Coloring powers of planar graphs [J]. SIAM J Discret Math 16(4):651–662

    Article  MATH  MathSciNet  Google Scholar 

  • Borodin OV, Broersma HJ, Glebov AN, van den Heuvel J (2001) The minimum degree and chromatic number of the square of a planar graph [J]. Diskret Anal Issled Oper 8(4):9–33

    MATH  Google Scholar 

  • Borodin OV, Ivanova AO, Neustroeva TK (2006) (p, q)-coloring of sparse planar graphs [J]. Mat Zametki YaGU 13(2):3–9

    Google Scholar 

  • Borodin OV, Ivanova AO, Neustroeva TK (2006) List (p, q)-coloring of sparse planar graphs [J]. Sibirsk Elektron Mat Izv 3:355–361

    MATH  MathSciNet  Google Scholar 

  • Borodin OV, Broersma HJ, Glebov AN, van den Heuvel J (2001) The structure of plane triangulations in terms of stars and bunches [J]. Diskret Anal Issled Oper 8(2):15–39

    MATH  Google Scholar 

  • Borodin OV, Ivanova AO (2009) 2-Distance (\(\Delta +2\))-coloring of planar graphs with girth six and \(\Delta \ge 18\) [J]. Discret Math 309:6496–6502

    Article  MATH  MathSciNet  Google Scholar 

  • Borodin OV, Ivanova AO, Neustroeva TK (2005) List 2-distance(\(\Delta +1\))-coloring of planar graphs with given girth. Diskret Anal Issled Oper 12(3):32–47

    MATH  MathSciNet  Google Scholar 

  • Bonamy Marthe (2013) Graphs with maximum degree \(\Delta \ge 17\) and maximum average degree less than 3 are list 2-distance (\(\Delta +2\))-colorable [J]. Discret Math 313:427–449

    Google Scholar 

  • Borodin OV, Ivanova AO (2009) List 2-distance (\(\Delta +2\))-coloring of planar graphs with girth six and \(\Delta \ge 24\). Sib Math J 50(6):958–964

    Article  MathSciNet  Google Scholar 

  • Cranston DW, Erman R, Skrekovski R (2013) Choosability of the square of a planar graph with maximum degree four, manuscript

  • Dvorak Z, Kral D, Nejedly P et al (2005) Coloring squares of planar graphs with no short cycles [J]. Discret Appl Math 43:976–1008

    Google Scholar 

  • Molloy M, Salavatipour MR (2012) Frequency channel assignment on planar networks [J], algorithms-ESA 2002. Springer, Berlin, pp 736–747

    Google Scholar 

  • Wang FW, Lih W (2003) Labeling planar graphs with conditions on girth and distance two. SIAM J Discret Math 17(2):264–275

    Article  MATH  MathSciNet  Google Scholar 

  • Wegner G (1977) Graphs with given diameter and a coloring problem [R]. University of Dortmund, Berlin

    Google Scholar 

  • YueHua Bu, Zhu Xubo (2012) An optimal square coloring of planar graphs. J Comb Optim 24:580–592

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

Research supported partially by NSFC (No. 11271334) and ZJNSF (No. Z6110786).

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Authors and Affiliations

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

    Yuehua Bu & Xiaoyan Yan

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  1. Yuehua Bu
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  2. Xiaoyan Yan
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Correspondence to Yuehua Bu.

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Bu, Y., Yan, X. List 2-distance coloring of planar graphs. J Comb Optim 30, 1180–1195 (2015). https://doi.org/10.1007/s10878-013-9700-2

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  • DOI: https://doi.org/10.1007/s10878-013-9700-2

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