  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.
