Supported in part by USA-Israel BSF Grant (2006-322 and 2010-115); Israel Science Foundation (1063/08).
Article first published online: 28 MAY 2012
Copyright © 2012 Wiley Periodicals, Inc.
Random Structures & Algorithms
Volume 43, Issue 1, pages 1–15, August 2013
How to Cite
Krivelevich, M., Lubetzky, E. and Sudakov, B. (2013), Longest cycles in sparse random digraphs. Random Struct. Alg., 43: 1–15. doi: 10.1002/rsa.20435
Supported in part by NSF grant (DMS-1101185); USA-Israel BSF Grant.
- Issue published online: 20 JUN 2013
- Article first published online: 28 MAY 2012
- Manuscript Accepted: 30 MAR 2012
- Manuscript Received: 14 FEB 2011
- directed graphs;
- random graphs;
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 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 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 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. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013