Arbitrary Orientations Of Hamilton Cycles In Oriented Graphs

TL;DR

This paper proves that large oriented graphs with high minimum in-degree and out-degree contain all possible Hamilton cycle orientations, confirming a longstanding conjecture.

Contribution

It introduces a randomized embedding method to establish the existence of all Hamilton cycle orientations in large oriented graphs with specified degree conditions.

Findings

Large oriented graphs with minimum degrees above (3/8 + α)|G| contain all Hamilton cycle orientations.

The result confirms a conjecture by H"aggkvist and Thomason.

The method used is a novel randomized embedding approach.

Abstract

We use a randomised embedding method to prove that for all \alpha>0 any sufficiently large oriented graph G with minimum in-degree and out-degree \delta^+(G),\delta^-(G)\geq (3/8+\alpha)|G| contains every possible orientation of a Hamilton cycle. This confirms a conjecture of H\"aggkvist and Thomason.