Lower bound of the hadwiger number of graphs by their average degree


The aim of this paper is to show that the minimum Hadwiger number of graphs with average degreek isO(k/√logk). Specially, it follows that Hadwiger’s conjecture is true for almost all graphs withn vertices, furthermore ifk is large enough then for almost all graphs withn vertices andnk edges.

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  1. [1]

    P. Erdős, J. Spencer,Probabilistic methods in combinatorics, Academic Press, New York-London, 1974.

    Google Scholar 

  2. [2]

    H. Hadwiger, Über eine Klassifikation der Streckenkomplexe, Viert.Naturforsch. Ges. Zürich,88 (1943), 133–142.

    MathSciNet  Google Scholar 

  3. [3]

    A. D. Korshunov, On the chromatic number of graphs onn vertices (in Russian),Metody diskretnogo analiza v teorii bulevyh funkcij i skhem,35 (1980), 15–45.

    MATH  Google Scholar 

  4. [4]

    A. V. Kostochka, On the minimum Hadwiger number of graphs with given average degree (in Russian),submitted to Diskretnij Analiz.

  5. [5]

    W. Mader, Homomorphiesätze für Graphen,Math. Annalen,178 (1968), 154–168.

    MATH  Article  MathSciNet  Google Scholar 

  6. [6]

    Z. Miller, Contractions of graphs: A theorem of Ore and an extremal problem,Discrete Math.,21 (1978), 261–273.

    Article  MathSciNet  Google Scholar 

  7. [7]

    K. Wagner, Beweis einer Abschwächung der Hadwiger—Vermutung,Math. Annalen,153 (1964), 139–141.

    MATH  Article  Google Scholar 

  8. [8]

    B. Zelinka, On some graph-theoretical problems of V. G. Vizing,Cas. Pestov. Math.,98 (1973), 56–66.

    MathSciNet  Google Scholar 

  9. [9]

    A. A. Zykov, On the edge number of graphs with no greater Hadwiger number than 4,Prikladnaja matematika i programmirovanije,7 (1972), 52–55.

    MathSciNet  Google Scholar 

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Kostochka, A.V. Lower bound of the hadwiger number of graphs by their average degree. Combinatorica 4, 307–316 (1984). https://doi.org/10.1007/BF02579141

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