Hamilton Cycles in Dense Regular Digraphs and Oriented Graphs

TL;DR

This paper proves that dense regular oriented graphs with degree above a certain threshold contain Hamilton cycles, confirming an approximate version of Jackson's 1981 conjecture and addressing related conjectures on regular directed graphs.

Contribution

It establishes the existence of Hamilton cycles in dense regular oriented graphs with degree above (1/4 + ε)n, advancing understanding of Hamiltonicity in such graphs.

Findings

Regular oriented graphs with degree > (1/4 + ε)n have Hamilton cycles

Confirms an approximate version of Jackson's conjecture from 1981

Addresses conjectures on Hamiltonicity of regular directed graphs

Abstract

We prove that for every $\varepsilon > 0$ there exists $n_0=n_0(\varepsilon)$ such that every regular oriented graph on $n > n_0$ vertices and degree at least $(1/4 + \varepsilon)n$ has a Hamilton cycle. This establishes an approximate version of a conjecture of Jackson from 1981. We also establish a result related to a conjecture of K\"uhn and Osthus about the Hamiltonicity of regular directed graphs with suitable degree and connectivity conditions.