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The Chromatic Number of Graphs with No Induced Subdivision of \(K_4\)

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Abstract

In 2012, Lévêque, Maffray and Trotignon conjectured that if a graph does not contain an induced subdivision of \(K_4\), then it is 4-colorable. Recently, Le showed that every such graph is 24-colorable. In this paper, we improve the upper bound to 8.

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References

  1. Chudnovsky, M., Liu, C.-H., Schaudt, O., Spirkl, S., Trotignon, N., Vušković, K.: Triangle-free graphs that do not contain an induced subdivision of K4 are 3-colorable. J. Graph Theroy 92, 67–95 (2019)

    Article  Google Scholar 

  2. Gyárfás, A.: Problems from the world surrounding perfect graphs. Zastow Mat. Appl. Math. 19, 413–441 (1987)

    MathSciNet  MATH  Google Scholar 

  3. Kühn, D., Osthus, D.: Induced subdivisions in Ks,s-free graphs of large average degree. Combinatorica 24, 287–304 (2004)

    Article  MathSciNet  Google Scholar 

  4. Le, N.K.: Chromatic number of ISK4-free graphs. Graphs Comb. 33, 1635–1646 (2017)

    Article  MathSciNet  Google Scholar 

  5. Lévêque, B., Maffray, F., Trotignon, N.: On graphs with no induced subdivision of K4. J. Comb. Theory Ser. B 102, 924–947 (2012)

    Article  Google Scholar 

  6. Pawlik, A., Kozik, J., Krawczyk, T., Lasoń, M., Micek, P., Trotter, W.T., Walczak, B.: Triangle-free intersection graphs of line segments with large chromatic number. J. Comb. Theory Ser. B 105, 6–10 (2014)

    Article  MathSciNet  Google Scholar 

  7. Scott, A.D.: Induced trees in graphs of large chromatic number. J. Graph Theroy 24, 297–311 (1997)

    Article  MathSciNet  Google Scholar 

  8. Trotignon, N., Vušković, K.: On triangle-free graphs that do not contain a subdivision of the complete graph on four vertices as an induced subgraph. J. Graph Theroy 84, 233–248 (2017)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Georgia State University, Atlanta, GA, 30303, USA

    Guantao Chen

  2. Research Center of Nonlinear Science, School of Mathematics and Computer Science, Wuhan Textile University, Wuhan, 430073, People’s Republic of China

    Yuan Chen

  3. Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, People’s Republic of China

    Qing Cui

  4. Faculty of Science, Jiangxi University of Science and Technology, Ganzhou, 341000, People’s Republic of China

    Xing Feng

  5. Center for Discrete Mathematics, Fuzhou University, Fuzhou, 350108, People’s Republic of China

    Qinghai Liu

Authors
  1. Guantao Chen
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  2. Yuan Chen
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  3. Qing Cui
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  4. Xing Feng
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  5. Qinghai Liu
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Corresponding author

Correspondence to Yuan Chen.

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G. Chen: partially supported by the NSF Grant DMS-1855716. Y. Chen: partially supported by the National Natural Science Foundation of China (no. 11526160) and the Science and Technology Innovation Project of Wuhan Textile University. Q. Cui: partially supported by the National Natural Science Foundation of China (no. 11501291). X. Feng: partially supported by the Foundation of Jiangxi Provincial Education Department of China (no. GJJ190490). Q. Liu: partially supported by the National Natural Science Foundation of China (no. 11871015)

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Chen, G., Chen, Y., Cui, Q. et al. The Chromatic Number of Graphs with No Induced Subdivision of \(K_4\). Graphs and Combinatorics 36, 719–728 (2020). https://doi.org/10.1007/s00373-020-02148-x

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  • DOI: https://doi.org/10.1007/s00373-020-02148-x

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