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 ${\mathrm{deg}}^{+}(x)+{\mathrm{deg}}^{-}(y)\ge (3n-3)/4$ whenever $G$ does not have an edge from $x$ to $y$ contains a Hamilton cycle. This is best possible and solves a problem of Kühn and Osthus from 2012. Our result generalizes the result of Keevash, Kühn, and Osthus and improves the asymptotic bound obtained by Kelly, Kühn, and Osthus.