Robust Hamiltonicity of random directed graphs

• Published in: Journal of combinatorial theory. Series B 2017-09, Vol.126, p.1-23
• Main Authors: Ferber, Asaf, Nenadov, Rajko, Noever, Andreas, Peter, Ueli, Škorić, Nemanja
• Format: Article
• Language: English
• Subjects: Directed Hamilton cycle; Local resilience; Random digraphs
• ISSN: 0095-8956; 1096-0902
• DOI: 10.1016/j.jctb.2017.03.006

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 ⌊n/2⌋ 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=ω(log⁡n/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=ω(log⁡n/n). In this paper we significantly improve their result to p=ω(log8⁡n/n), which is optimal up to the polylogarithmic factor.