Independent sets ink-chromatic graphs

Combinatorica volume 5, pages261–269(1985)Cite this article

Abstract

A graph is said to have propertyP k if in eachk-colouring ofG using allk colours there arek independent vertices having all colours. An (unpublished) suggestion of P. Erdős is answered in the affirmative: For eachk≧3 there is a k-critical graph withP k . With the aid of a construction of T. Gallaik-chromatic graphs (k≧7) withP k orP k+1 of arbitrarily high connectivity are obtained. The main result is: Eachk-chromatic graph (k≧3) of girth ≧6 hasP k or is a circuit of length 7.

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References

  1. [1]

    W. G. Brown andJ. W. Moon, Sur les ensembles de sommets, indépendant dans les graphs chromatiques minimaux,Canad. J. Math. 21 (1969), 274–278.

    MATH  MathSciNet  Google Scholar 

  2. [2]

    G. A. Dirac, Circuits in critical graphs,Monatsh. Math. 59 (1955), 178–187.

    MATH  Article  MathSciNet  Google Scholar 

  3. [3]

    P. Erdős, Graph theory and probability,Canad. J. Math. 11 (1959), 34–38.

    MathSciNet  Google Scholar 

  4. [4]

    U. Krusenstjerna-Hafstrøm andB. Toft, Some remarks on Hadwiger’s conjecture and its relation to a conjecture of Lovász, in:The Theory and Applications of Graphs. (Ed. G. Chartrand), 1981, John Wiley Inc., 449–459.

  5. [5]

    L. Lovász, On chromatic number of finite set systems,Acta Math. Acad. Sci. Hung. 19 (1968), 59–67.

    MATH  Article  Google Scholar 

  6. [6]

    L. Lovász, Independent sets in critical chromatic graphs,Studia Sci. Math. Hung. 8 (1973), 165–168.

    Google Scholar 

  7. [7]

    O. Ore,The Four Color Problem, Academic Press, 1967.

  8. [8]

    V. Rödl andZ. Tuza, On colour critical graphs,submitted to Combin. Theory B.

  9. [9]

    H. Sachs, „Einführung in die Theorie der endlichen Graphen“, Teil I, Teubner, Leipzig, 1970).

    Google Scholar 

  10. [10]

    M. Simonovits, On colour-critical graphs,Studia Sci. Math. Hung. 7 (1972), 67–81.

    MATH  MathSciNet  Google Scholar 

  11. [11]

    H.-J. Voss, Independent sets in (k+1)-colorations ofk-chromatic graphs,Wiss. Z. TH Ilmenau 30 (1984), 27–40.

    MATH  MathSciNet  Google Scholar 

  12. [12]

    H.-J. Voss, Unabhängige Mengen ink-chromatischen Graphen,Tagungsberichte des 2. Kol. über Geometrie und Kombinatorik, Karl-Marx-Stadt, 13–14. 10. 1983.

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Affiliations

  1. Pädagogische Hochschule „Karla Friedrich Wilhelm Wander“ Dresden, Wigard str. 17 PSF 365, 8060, Dresden, German Democratic Republic

    H. -J. Voss

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  1. H. -J. Voss
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Dedicated to Paul Erdős on his seventieth birthday

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Voss, H.J. Independent sets ink-chromatic graphs. Combinatorica 5, 261–269 (1985). https://doi.org/10.1007/BF02579371

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AMS subject classification (1980)

  • 05 C 15