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/2 contains a directed Hamilton cycle. In this paper we extend this theorem to a random directed graph 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 ≫ log n/√n, then a.a.s. every subdigraph of D(n,p) with minimum out-degree and in-degree at least (1/2+o(1))np contains a directed Hamilton cycle. The constant 1/2 is asymptotically best possible. Our result also strengthens classical results about the existence of directed Hamilton cycles in random directed graphs.