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arXiv · 1404.4734

Random directed graphs are robustly Hamiltonian

Dan Hefetz·Angelika Steger·Benny Sudakov
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Abstract

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 ${\mathcal D}(n,p)$, 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 $p \gg \log n/\sqrt{n}$, then a.a.s. every subdigraph of ${\mathcal D}(n,p)$ with minimum out-degree and in-degree at least $(1/2 + o(1)) n p$ 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.

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Dan Hefetz, Angelika Steger, Benny Sudakov. 2014-04-18. Random directed graphs are robustly Hamiltonian. https://arxiv.org/abs/1404.4734

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