Advertisement

Combinatorica

, Volume 4, Issue 4, pp 307–316 | Cite as

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

  • A. V. Kostochka
Article

Abstract

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.

AMS subject classification 1980)

05 C 10 05 C 15 60 C 05 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  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.MathSciNetGoogle 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.zbMATHGoogle Scholar
  4. [4]
    A. V. Kostochka, On the minimum Hadwiger number of graphs with given average degree (in Russian),submitted to Diskretnij Analiz.Google Scholar
  5. [5]
    W. Mader, Homomorphiesätze für Graphen,Math. Annalen,178 (1968), 154–168.zbMATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    Z. Miller, Contractions of graphs: A theorem of Ore and an extremal problem,Discrete Math.,21 (1978), 261–273.CrossRefMathSciNetGoogle Scholar
  7. [7]
    K. Wagner, Beweis einer Abschwächung der Hadwiger—Vermutung,Math. Annalen,153 (1964), 139–141.zbMATHCrossRefGoogle Scholar
  8. [8]
    B. Zelinka, On some graph-theoretical problems of V. G. Vizing,Cas. Pestov. Math.,98 (1973), 56–66.MathSciNetGoogle 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.MathSciNetGoogle Scholar

Copyright information

© Akadémiai Kiadó 1984

Authors and Affiliations

  • A. V. Kostochka
    • 1
  1. 1.Mathematical Institute of the Siberian Branch of theSoviet Academy of SciencesU.S.S.R.

Personalised recommendations