The chromatic number of random graphs at the double-jump threshold

Abstract

A crucial step in the Erdös-Rényi (1960) proof that the double-jump threshold is also the planarity threshold for random graphs is shown to be invalid. We prove that whenp=1/n, almost all graphs do not contain a cycle with a diagonal edge, contradicting Theorem 8a of Erdös and Rényi (1960). As a consequence, it is proved that the chromatic number is 3 for almost all graphs whenp=1/n.

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Research supported U.S. National Science Foundation Grants DMS-8303238 and DMS-8403646. The research was conducted on an exchange visit by Professor Wierman to Poland supported by the national Academy of Sciences of the USA and the Polish Academy of Sciences.

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Łuczak, T., Wierman, J.C. The chromatic number of random graphs at the double-jump threshold. Combinatorica 9, 39–49 (1989). https://doi.org/10.1007/BF02122682

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AMS Subject Classification (1980)

  • Primary 05 C 80
  • Secondary 05 C 10
  • 05 C 15
  • 05 C 3o