Almost all graphs with high girth and suitable density have high chromatic number

  • Authors:

  • Deryk Osthus;Hans Jürgen Prömel;Anusch Taraz

  • Affiliations:

  • (Graduate school “Algorithmische Diskrete Mathematik.”) Institut für Informatik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany;Institut für Informatik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany;Institut für Informatik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany

  • Venue:
  • Journal of Graph Theory
  • Year:
  • 2001

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Abstract

Erdös proved that there exist graphs of arbitrarily highgirth and arbitrarily high chromatic number. We give a differentproof (but also using the probabilistic method) that also yieldsthe following result on the typical asymptotic structure of graphsof high girth: for all 𝓁 ≥ 3 and k εℕ there exist constants C1 andC2 so that almost all graphs on n verticesand m edges whose girth is greater than 𝓁 havechromatic number at least k, provided that $C_{1}n\,\leq\,m\,\leq\, C_{2}n^{\ell /(\ell -1)}$. © 2001 John Wiley &Sons, Inc. J Graph Theory 37: 220226, 2001