Longest cycles in sparse random digraphs
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 + ?)ce?cn vertices with high probability, where ? = ? (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 + ?)e?cn vertices w.h.p., where ? = ? (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
cycles random graphs directed graphs