The longest path in a random graph

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

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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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AMS subject classification (1980)

  • 05 C 38
  • 60 C 05
  • 60 J 80