An exact Ore-degree condition for Hamilton cycles in oriented graphs

Abstract

An oriented graph is a digraph that contains no 2-cycles, i.e., there is at most one arc between any two vertices. We show that every oriented graph $G$ of sufficiently large order $n$ with $\mathrm{deg}^+(x) +\mathrm{deg}^{-}(y)\geq (3n-3)/4$ whenever $G$ does not have an edge from $x$ to $y$ contains a Hamilton cycle. This is best possible and solves a problem of K\"uhn and Osthus from 2012. Our result generalizes the result of Keevash, K\"uhn, and Osthus and improves the asymptotic bound obtained by Kelly, K\"uhn, and Osthus.