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


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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  1. [1]

    M. Ajtai, J. Komlós andE. Szemerédi (1979), Topological complete subgraphs in random graphs,Studia Scientiarum Mathematicarum Hungarica 14, 293–297.

    Google Scholar 

  2. [2]

    M. Ajtai, J. Komlós andE. Szemerédi (1981), The longest path in a random graph,Combinatorica 1, 1–12.

    Google Scholar 

  3. [3]

    B.Bollobás (1984), The evolution of sparse graphs,Graph Theory and Combinatorics, Proc. Cambridge Combinatorial Conf. in honour of Paul Erdös (B. Bollobás, ed.), Academic Press, 35–57.

  4. [4]

    B.Bollobás (1985),Random Graphs. Academic Press.

  5. [5]

    A. Cayley (1889), A theorem on trees.Quart. J. Pure Appl. Math. 23, 376–378, orMath. Papers 13, 26–28.

    Google Scholar 

  6. [6]

    P. Erdös andA. Rényi (1960), On the evolution of random graphs,Magyar Tud. Akad. Mat. Kutató Int. Közl. 5, 17–61.

    Google Scholar 

  7. [7]

    L. Katz (1955), The probability of indecomposability of a random mapping function,Ann. Math. Stat. 26, 512–517.

    Google Scholar 

  8. [8]

    E.Palmer (1985),Graphical Evolution: An Introduction to the Theory of Random Graphs, John Wiley and Sons.

  9. [9]

    A. Rényi (1959), On connected graphs I,Magyar Tud. Akad. Mat. Kutató Int. Közl. 4, 385–388.

    Google Scholar 

  10. [10]

    H. J. Voss (1982), Graphs having circuits with at least two chords,J. Comb. Theory Ser. B 32, 264–285.

    Article  Google Scholar 

  11. [11]

    E. M. Wright (1980), The number of connected sparsely edged graphs III. Asymptotic results.J. Graph Theory 4, 393–407.

    Google Scholar 

Download references

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Additional information

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).

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AMS Subject Classification (1980)

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