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1-planar graphs with girth at least 6 are (1,1,1,1)-colorable

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Abstract

A graph is 1-planar if it can be drawn in the Euclidean plane so that each edge is crossed by at most one other edge. A 1-planar graph on n vertices is optimal if it has 4n − 8 edges. We prove that 1-planar graphs with girth at least 6 are (1,1,1,1)-colorable (in the sense that each of the four color classes induces a subgraph of maximum degree one). Inspired by the decomposition of 1-planar graphs, we conjecture that every 1-planar graph is (2,2,2,0,0)-colorable.

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Acknowledgments

We would like to thank the anonymous reviewers for their valuable comments and helpful suggestions.

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Authors and Affiliations

  1. School of Computer Science and Technology, Shandong Jianzhu University, Jinan, 250358, P. R. China

    Lili Song

  2. Department of Mathematics and Statistics, Shandong Normal University, Fengming Road, Lingang Development Zone, Jinan, 250358, P. R. China

    Lili Song

  3. Department of Mathematics and Statistics, Shandong Normal University, No. 88 East Wenhua Road, Lixia District, Jinan, 250358, P. R. China

    Lei Sun

Authors
  1. Lili Song
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  2. Lei Sun
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Correspondence to Lei Sun.

Additional information

The research has been supported by the National Natural Science Foundation of China (Grant No. 11701342 and Grant No. 12271311) and the Natural Science Foundation of Shandong Province (Grant No. ZR2016AQ01) of China.

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Song, L., Sun, L. 1-planar graphs with girth at least 6 are (1,1,1,1)-colorable. Czech Math J 73, 993–1006 (2023). https://doi.org/10.21136/CMJ.2023.0418-21

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  • DOI: https://doi.org/10.21136/CMJ.2023.0418-21

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