# Longest cycles in sparse random digraphs

Michael Krivelevich, Eyal Lubetzky, Benny Sudakov

Research output: Contribution to journalArticle

### 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 language English (US) 1-15 15 Random Structures and Algorithms 43 1 https://doi.org/10.1002/rsa.20435 Published - Aug 2013

Long Cycle
Digraph
Random Graphs
Cycle
Longest Path
Sparse Graphs
Estimate
Lower bound
Zero

### Keywords

• Cycles
• Directed graphs
• Random graphs

### ASJC Scopus subject areas

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

### Cite this

Longest cycles in sparse random digraphs. / Krivelevich, Michael; Lubetzky, Eyal; Sudakov, Benny.

In: Random Structures and Algorithms, Vol. 43, No. 1, 08.2013, p. 1-15.

Research output: Contribution to journalArticle

Krivelevich, Michael ; Lubetzky, Eyal ; Sudakov, Benny. / Longest cycles in sparse random digraphs. In: Random Structures and Algorithms. 2013 ; Vol. 43, No. 1. pp. 1-15.
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