Longest cycles in sparse random digraphs

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 $\cG(n,p)$ with $p=c/n$ has a cycle on at all but at most $(1+\epsilon)ce^{-c}n$ vertices with high probability, where $\epsilon=\epsilon(c)\to 0$ as $c\to\infty$. This estimate on the number of uncovered vertices is essentially tight due to vertices of degree 1. However, for the random digraph $\cD(n,p)$ 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 $\cD(n,p)$ with $p=c/n$ has a cycle containing all but $(2+\epsilon)e^{-c}n$ vertices w.h.p., where $\epsilon=\epsilon(c)\to 0$ as $c\to\infty$. 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.