Random directed graphs are robustly Hamiltonian

A classical theorem of Ghouila-Houri from 1960 asserts that every directed graph on $n$ vertices with minimum out-degree and in-degree at least $n\mathrm{/}2$ contains a directed Hamilton cycle. In this paper we extend this theorem to a random directed graph $\mathcal{D}(n,p)$, that is, a directed graph in which every ordered pair $(u,v)$ becomes an arc with probability $p$ independently of all other pairs. Motivated by the study of resilience of properties of random graphs, we prove that if $p\gg \log n\mathrm{/}\sqrt{n}$, then a.a.s. every subdigraph of $\mathcal{D}(n,p)$ with minimum out-degree and in-degree at least $(1/2 + o(1)) n p$containsadirectedHamiltoncycle\mathrm{.}Theconstant$1/2$ is asymptotically best possible. Our result also strengthens classical results about the existence of directed Hamilton cycles in random directed graphs.