Arbitrary orientations of Hamilton cycles in directed graphs of large minimum degree

TL;DR

This paper extends classical results on Hamilton cycles in directed graphs by showing that large minimum degree guarantees the presence of all cycle orientations, except for the directed cycle in non-strongly connected graphs.

Contribution

It generalizes Ghouila-Houri's theorem by proving that high minimum degree ensures all cycle orientations are present in directed graphs, barring the directed cycle in non-strongly connected cases.

Findings

High minimum degree guarantees all cycle orientations in directed graphs.

Almost all orientations of Hamilton cycles are contained, except for the directed cycle in non-strongly connected graphs.

The minimum degree condition is asymptotically optimal.

Abstract

In 1960, Ghouila-Houri proved that every strongly connected directed graph $G$ on $n$ vertices with minimum degree at least $n$ contains a directed Hamilton cycle. We asymptotically generalize this result by proving the following: every directed graph $G$ on $n$ vertices and with minimum degree at least $(1+o(1))n$ contains every orientation of a Hamilton cycle, except for the directed Hamilton cycle in the case when $G$ is not strongly connected. In fact, this minimum degree condition forces every orientation of a cycle in $G$ of every possible length, other than perhaps the directed cycles.