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On a Coloring Conjecture of Hajós

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Abstract

Hajós conjectured that graphs containing no subdivision of \(K_5\) are 4-colorable. It is shown in Yu and Zickfeld (J Comb Theory Ser B 96:482–492, 2006) that if there is a counterexample to this conjecture then any minimum such counterexample must be 4-connected. In this paper, we further show that if \(G\) is a minimum counterexample to Hajós’ conjecture and \(S\) is a 4-cut in \(G\) then \(G-S\) has exactly two components.

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References

  1. Catlin, P.: Hajós’ graph-coloring conjecture: variations and counterexamples. J. Comb. Theory Ser. B 26, 268–274 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  2. Dirac, G.A.: A property of 4-chromatic graphs and some remarks on critical graphs. J. Lond. Math. Soc. Ser. B 27, 85–92 (1952)

    Article  MATH  MathSciNet  Google Scholar 

  3. Erdös, P., Fajtlowicz, S.: On the conjecture of Hajós. Combinatorica 1, 141–143 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  4. Kelmans, A.K.: Every minimal counterexample to the Dirac conjecture is 5-connected. Lectures to the Moscow Seminar on Discrete Mathematics, (1979)

  5. Ma, J., Yu, X.: Independent paths and \(K_5\)-subdivisions. J. Comb. Theory Ser. B 100, 600–616 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  6. Ma, J., Yu, X.: \(K_5\)-Subdivisions in graphs containing \(K_4^-\). J. Comb. Theory Ser. B 103, 713–732 (2013)

    Article  MathSciNet  Google Scholar 

  7. Mader, W.: \(3n-5\) Edges do force a subdivision of \(K_5\). Combinatorica 18, 569–595 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  8. Perfect, H.: Applications of Menger’s graph theorem. J. Math. Anal. Appl. 22, 96–111 (1968)

    Article  MATH  MathSciNet  Google Scholar 

  9. Robertson, N., Chakravarti, K.: Covering three edges with a bond in a nonseparable graph. In: Deza, M., Rosenberg, G. (eds.) Combinatorics 79 Part I. Annals of discrete mathematics, p. 247 (1980)

  10. Seymour, P.D.: Private communication

  11. Seymour, P.D.: Disjoint paths in graphs. Discret. Math. 29, 293–309 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  12. Shiloach, Y.: A polynomial solution to the undirected two paths problem. J. Assoc. Comp. Mach. 27, 445–456 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  13. Thomassen, C.: 2-Linked graphs. Eur. J. Comb. 1, 371–378 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  14. Watkins, M.E., Mesner, D.M.: Cycles and connectivity in graphs. Can. J. Math. 19, 1319–1328 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  15. Yu, X., Zickfeld, F.: Reducing Hajos’ coloring conjecture to 4-connected graphs. J. Comb. Theory Ser. B 96, 482–492 (2006)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgments

We thank an anonymous referee for discovering a missing case in the original manuscript, and for an observation that led to a more direct argument in Case (4.1) in Sect. 3.

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Authors and Affiliations

  1. School of Mathematics and Physics, Shanghai University of Electric Power, Shanghai, 200090, China

    Yuqin Sun

  2. School of Mathematics, Georgia Institute of Technology, Atlanta, GA, 30332-0160, USA

    Xingxing Yu

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  1. Yuqin Sun
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  2. Xingxing Yu
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Correspondence to Xingxing Yu.

Additional information

Partially supported by NSF Grants DMS-1265564 and AST-1247545 and NSA Grant H98230-13-1-0255.

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Sun, Y., Yu, X. On a Coloring Conjecture of Hajós. Graphs and Combinatorics 32, 351–361 (2016). https://doi.org/10.1007/s00373-015-1539-0

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  • DOI: https://doi.org/10.1007/s00373-015-1539-0

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