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Borodin–Kostochka’s conjecture on \((P_5,C_4)\)-free graphs

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Abstract

Brooks’ theorem states that for a graph G, if \(\varDelta (G)\ge 3\), then \(\chi (G)\le \max \{\varDelta (G),\omega (G)\}\). Borodin and Kostochka conjectured a result strengthening Brooks’ theorem, stated as, if \(\varDelta (G)\ge 9\), then \(\chi (G)\le \max \{\varDelta (G)-1,\omega (G)\}\). This conjecture is still open for general graphs. In this paper, we show that the conjecture is true for graphs having no induced path on five vertices and no induced cycle on four vertices.

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References

  1. Brooks, R.L.: On colouring the nodes of a network. In: Mathematical Proceedings of the Cambridge Philosophical Society. Cambridge University Press, Cambridge, vol. 37, issue 2, pp. 194–197 (1941)

  2. Borodin, O.V., Kostochka, A.V.: On an upper bound of a graph’s chromatic number, depending on the graph’s degree and density. J. Comb. Theory Ser. B 23(2–3), 247–250 (1977)

    Article  MathSciNet  Google Scholar 

  3. Catlin, P.A.:. Embedding subgraphs and coloring graphs under extremal degree conditions. ProQuest LLC, Ann Arbor, MI, Thesis (Ph.D.). The Ohio State University (1976)

  4. Cranston, D.W., Lafayette, H., Rabern, L.: Coloring \((P_5,gem)\)-free graphs with \(\Delta -1\) colors. arXiv:2006.02015 (2020)

  5. Cranston, D.W., Rabern, L.: Graphs with \(\chi =\Delta \) have big cliques. SIAM J. Discrete Math. 29(4), 1792–1814 (2015)

    Article  MathSciNet  Google Scholar 

  6. Cranston, D.W., Rabern, L.: Coloring claw-free graphs with \(\Delta -1\) colors. SIAM J. Discrete Math. 27(1), 534–549 (2013)

    Article  MathSciNet  Google Scholar 

  7. Dirac, G.A.: Note on the colouring of graphs. Math. Z. 54, 347–353 (1951)

    Article  MathSciNet  Google Scholar 

  8. Dhurandhar, M.: Improvement on Brooks’ chromatic bound for a class of graphs. Discrete Math. 42(1), 51–56 (1982)

    Article  MathSciNet  Google Scholar 

  9. Jensen, T.R., Toft, B.: Graph Coloring Problems. Wiley, New York (1995)

    MATH  Google Scholar 

  10. Kierstead, H.A., Schmerl, J.H.: The chromatic number of graphs which induce neither \(K_{1,3}\) nor \(K_5-e\). Discrete Math. 58(3), 253–262 (1986)

    Article  MathSciNet  Google Scholar 

  11. Kohl, A., Schiemeyer, I.: Some results on Reed’s Conjecture about \(\omega \), \(\Delta \), and \(\chi \) with respect to \(\alpha \). Discrete Math. 310(9), 1429–1438 (2010)

    Article  MathSciNet  Google Scholar 

  12. Kostochka, A.V.: Degree, density, and chromatic number. Metody Diskret. Anal. 35, 45–70 (1980)

    MathSciNet  MATH  Google Scholar 

  13. Kostochka, A.V., Rabern, L., Stiebitz, M.: Graphs with chromatic number close to maximum degree. Discrete Math. 312(6), 1273–1281 (2012)

    Article  MathSciNet  Google Scholar 

  14. Reed, B.: A strengthening of Brooks’ theorem. J. Comb. Theory Ser. B 76(2), 136–149 (1999)

    Article  MathSciNet  Google Scholar 

  15. Schiermeyer, I., Randerath, B.: Polynomial \(\chi \)-binding functions and forbidden induced subgraphs—a survey. Graphs Combin. 35, 1–31 (2018)

    Article  MathSciNet  Google Scholar 

  16. Scott, A., Seymour, P.: A survey of \(\chi \)-boundedness. arXiv:1812.07500 (2018)

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Authors and Affiliations

  1. Department of Mathematics and Computing, Indian Institute of Technology (ISM), Dhanbad, India

    Uttam K. Gupta & D. Pradhan

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  1. Uttam K. Gupta
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  2. D. Pradhan
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Correspondence to D. Pradhan.

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Gupta, U.K., Pradhan, D. Borodin–Kostochka’s conjecture on \((P_5,C_4)\)-free graphs. J. Appl. Math. Comput. 65, 877–884 (2021). https://doi.org/10.1007/s12190-020-01419-3

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  • DOI: https://doi.org/10.1007/s12190-020-01419-3

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