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Author(s)

Krivelevich Michael,

Lubetzky Eyal,

Sudakov Benny

Publication year2013

Publication title

random structures and algorithms

Resource typeJournals

PublisherWiley Subscription Services

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 (2 e − c − o (1)) n vertices with zero in‐degree or out‐degree. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013

Subject(s)acoustics , combinatorics , degree (music) , digraph , graph , mathematical analysis , mathematics , physics , random graph , upper and lower bounds

Language(s)English

SCImago Journal Rank1.314

H-Index69

eISSN1098-2418

pISSN1042-9832

DOI10.1002/rsa.20435

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