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The longest path in a random graph

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Abstract

A random graph with (1+ε)n/2 edges contains a path of lengthcn. A random directed graph with (1+ε)n edges contains a directed path of lengthcn. This settles a conjecture of Erdõs.

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References

  1. P. Erdős andA. Rényi, On the evolution of random graphs,Publications of the Math. Inst. the Hung. Acad. of Sci.,5 (1960), 17–61.

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  2. A. Rényi, Some remarks on the theory of trees,Publications of the Math. Inst. of the Hung. Acad. of Sci.,4 (1959), 73–85.

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  3. W. F. de la Vega, Sur la plus grande longueur des chemins élémentaires de graphes aléatoires,Preprint of Laboratoire d’informatique pour les Sciences de l’Homme, C.N.R.S., 1979.

  4. P. Erdős, Problems and results on finite and infinite graphs, inProc. Symp. Prague 1974, Akademia Praha 1975.

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Authors and Affiliations

  1. Mathematical Institute of the Hungarian Academy of Sciences, H-1053, Budapest, Hungary

    Miklós Ajtai, János Komlós & Endre Szemerédi

Authors
  1. Miklós Ajtai
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  2. János Komlós
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  3. Endre Szemerédi
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Ajtai, M., Komlós, J. & Szemerédi, E. The longest path in a random graph. Combinatorica 1, 1–12 (1981). https://doi.org/10.1007/BF02579172

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  • DOI: https://doi.org/10.1007/BF02579172

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