Arbitrary Orientations of Hamilton Cycles in Digraphs

TL;DR

This paper proves that large digraphs with minimum in- and outdegree at least half the number of vertices contain all Hamilton cycle orientations except possibly the antidirected one, refining previous bounds.

Contribution

It establishes the existence of all Hamilton cycle orientations in such digraphs, except the antidirected case, with optimal degree conditions, improving prior approximate results.

Findings

Contains all Hamilton cycle orientations except possibly antidirected

Minimum degree threshold is tight at n/2

Improves upon previous approximate bounds

Abstract

Let $n$ be sufficiently large and suppose that $G$ is a digraph on $n$ vertices where every vertex has in- and outdegree at least $n/2$. We show that $G$ contains every orientation of a Hamilton cycle except, possibly, the antidirected one. The antidirected case was settled by DeBiasio and Molla, where the threshold is $n/2+1$. Our result is best possible and improves on an approximate result by H\"aggkvist and Thomason.