Hamiltonicity in random directed graphs is born resilient

Let {D-M}(M >= 0) be the n-vertex random directed graph process, where D-0 is the empty directed graph on n vertices, and subsequent directed graphs in the sequence are obtained by the addition of a new directed edge uniformly at random. For each epsilon > 0, we show that, almost surely, any directed graph D-M with minimum in- and out-degree at least 1 is not only Hamiltonian (as shown by Frieze), but remains Hamiltonian when edges are removed, as long as at most 1/2 - epsilon of both the in- and out-edges incident to each vertex are removed. We say such a directed graph is (1/2 - epsilon)-resiliently Hamiltonian. Furthermore, for each epsilon > 0, we show that, almost surely, each directed graph D-M in the sequence is not (1/2 + epsilon)-resiliently Hamiltonian. This improves a result of Ferber, Nenadov, Noever, Peter and Skoric, who showed, for each epsilon > 0, that the binomial random directed graph D(n, p) is almost surely (1/2 - epsilon)-resiliently Hamiltonian if p = omega(log(8) n/n).