Robust Hamiltonicity of random directed graphs

In his seminal paper from 1952 Dirac showed that the complete graph on n >= 3 vertices remains Hamiltonian even if we allow an adversary to remove left perpendicular n/2 right perpendicular edges touching each vertex. In 1960 Ghouila-Houri obtained an analogue statement for digraphs by showing that every directed graph on n >= 3 vertices with minimum in- and out-degree at least n/2 contains a directed Hamilton cycle. Both statements quantify the robustness of complete graphs (digraphs) with respect to the property of containing a Hamilton cycle. A natural way to generalize such results to arbitrary graphs (digraphs) is using the notion of local resilience. The local resilience of a graph (digraph) G with respect to a property P is the maximum number r such that G has the property P even if we allow an adversary to remove an r-fraction of (in and out-going) edges touching each vertex. The theorems of Dirac and Ghouila-Houri state that the local resilience of the complete graph and digraph with respect to Hamiltonicity is 1/2. Recently, this statements have been generalized to random settings. Lee and Sudakov (2012) proved that the local resilience of a random graph with edge probability p = omega (logn/n) with respect to Hamiltonicity is 1/2 +/- o(1). For random directed graphs, Hefetz, Steger and Sudakov (2014) proved an analogue statement, but only for edge probability p = w (log n/root n). In this paper we significantly improve their result to p = omega (log(8)n/n), which is optimal up to the poly logarithmic factor. (C) 2017 Elsevier Inc. All rights reserved.