### 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^{-c}n 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^{-c}n 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) |
---|---|

Pages (from-to) | 1-15 |

Number of pages | 15 |

Journal | Random Structures and Algorithms |

Volume | 43 |

Issue number | 1 |

DOIs | |

State | Published - Aug 2013 |

### Fingerprint

### Keywords

- Cycles
- Directed graphs
- Random graphs

### ASJC Scopus subject areas

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

### Cite this

*Random Structures and Algorithms*,

*43*(1), 1-15. https://doi.org/10.1002/rsa.20435

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

Research output: Contribution to journal › Article

*Random Structures and Algorithms*, vol. 43, no. 1, pp. 1-15. https://doi.org/10.1002/rsa.20435

}

TY - JOUR

T1 - Longest cycles in sparse random digraphs

AU - Krivelevich, Michael

AU - Lubetzky, Eyal

AU - Sudakov, Benny

PY - 2013/8

Y1 - 2013/8

N2 - 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.

AB - 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.

KW - Cycles

KW - Directed graphs

KW - Random graphs

UR - http://www.scopus.com/inward/record.url?scp=84879507203&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84879507203&partnerID=8YFLogxK

U2 - 10.1002/rsa.20435

DO - 10.1002/rsa.20435

M3 - Article

AN - SCOPUS:84879507203

VL - 43

SP - 1

EP - 15

JO - Random Structures and Algorithms

JF - Random Structures and Algorithms

SN - 1042-9832

IS - 1

ER -