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

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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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References

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

    Google Scholar 

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

    Google Scholar 

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

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

    Google Scholar 

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

    Google Scholar 

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

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

    Google Scholar 

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

    Google Scholar 

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Author information

Authors and Affiliations

  1. Institute of Mathematics, Adam Mickiewicz University, Poznan, Poland

    T. Łuczak

  2. Department of Mathematical Sciences, John Hopkins University, Baltimore, Maryland, USA

    J. C. Wierman

Authors
  1. T. Łuczak
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  2. J. C. Wierman
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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). https://doi.org/10.1007/BF02122682

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  • DOI: https://doi.org/10.1007/BF02122682

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