Longest cycles in sparse random digraphs

Michael Krivelevich, Eyal Lubetzky, Benny Sudakov

Research output: Contribution to journalArticlepeer-review

Abstract

Long paths and cycles in sparse random graphs and digraphs were studied intensively in the 1980's. It was finally shown by Frieze in 1986 that the random graph \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}${\mathcal{G}}(n,p)$\end{document} with p = c/n has a cycle on at all but at most (1 + ε)ce-cn vertices with high probability, where ε = ε (c) → 0 as c → ∞. This estimate on the number of uncovered vertices is essentially tight due to vertices of degree 1. However, for the random digraph \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}${\mathcal{D}}(n,p)$\end{document} no tight result was known and the best estimate was a factor of c/2 away from the corresponding lower bound. In this work we close this gap and show that the random digraph \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}${\mathcal{D}}(n,p)$\end{document} with p = c/n has a cycle containing all but (2 + ε)e-cn vertices w.h.p., where ε = ε (c) → 0 as c → ∞. This is essentially tight since w.h.p. such a random digraph contains (2e-c - o(1))n vertices with zero in-degree or out-degree.

Original languageEnglish (US)
Pages (from-to)1-15
Number of pages15
JournalRandom Structures and Algorithms
Volume43
Issue number1
DOIs
StatePublished - Aug 2013

Keywords

  • Cycles
  • Directed graphs
  • Random graphs

ASJC Scopus subject areas

  • Software
  • Mathematics(all)
  • Computer Graphics and Computer-Aided Design
  • Applied Mathematics

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