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

TL;DR

This paper establishes a precise degree condition in large oriented graphs that guarantees the existence of Hamilton cycles, solving a longstanding problem and improving previous bounds.

Contribution

It provides an exact Ore-degree condition for Hamilton cycles in large oriented graphs, confirming a conjecture and extending prior results.

Findings

Proves the degree condition is best possible.

Solves a problem posed by K"uhn and Osthus in 2012.

Generalizes previous results by Keevash, K"uhn, and Osthus.

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.