On the conjecture of hajós

Abstract

Hajós conjectured that everys-chromatic graph contains a subdivision ofK s, the complete graph ons vertices. Catlin disproved this conjecture. We prove that almost all graphs are counter-examles in a very strong sense.

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References

  1. [1]

    K. E. Appel andW. Haken, The existence of unavoidable sets of geographically good configurations,Illinois Journal of Mathematics, vol.20, 218–297.

  2. [2]

    P. Catlin, Hajós graph-colouring conjecture: variatons and counterexamples,Journal of Combinatorial Theory, Series B (to appear in 1979).

  3. [3]

    G. A. Dirac, A property of 4-chromatic graphs and some remarks on critical graphs,Journal of London Mathematical Society,27 (1952), 85–92.

    MATH  Article  MathSciNet  Google Scholar 

  4. [4]

    P. Erdős, Some extremal problems on families of graphs and related problems, Springer-Verlag,Lecture Notes in Mathematics,686, 13–21.

  5. [5]

    P. Erdős, Some remarks on chromatic graphs,Coll. Math. XVI (1967) 103–106.

    Google Scholar 

  6. [6]

    P. Erdős, Some remarks on graph theory,Bulletin of American Mathematical Society,53 (1947), 292–299.

    Article  Google Scholar 

  7. [7]

    P. Erdős andA. Hajnal, On complete topological subaraphs of certain graphs,Ann. Univ. Sci. Budapest,7 (1969), 193–199.

    Google Scholar 

  8. [8]

    S. Fajtlowicz, On the size independent sets in graphs,Proceedings of the IX Southeastern Conference on Combinatorics, Graph Theory and Computing, Utilitas Math, Congress. Num. XXI, (1978) 269–274.

  9. [9]

    H. Hadwiger, Über eine Klassifikationen der Streckenkomplexe,Vierteljschr. Naturforsch. Ges. Zürich,88 (1943), 133–143.

    MathSciNet  Google Scholar 

  10. [10]

    G. Hajós, Über eine Konstruktion nichtn-Farbbarer Graphen,Wiss. Z. Martin Luther Univ. Halle — Wittenberg Math. Naturwiss. Reihe 10 (1961), 116–117.

    Google Scholar 

  11. [11]

    W. Tutte, Colouring Problems,Mathematical Intelligencer 1 (1978), 72–75.

    MATH  MathSciNet  Article  Google Scholar 

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Erdős, P., Fajtlowicz, S. On the conjecture of hajós. Combinatorica 1, 141–143 (1981). https://doi.org/10.1007/BF02579269

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AMS (1980) subject classification

  • 05 C 15
  • 60 C 05