Hamilton cycles in dense regular digraphs and oriented graphs

Abstract

We prove that for every ε>0 there exists n0=n0(ε) such that every regular oriented graph on n>n0 vertices and degree at least (1/4+ε)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 Kühn and Osthus about the Hamiltonicity of regular directed graphs with suitable degree and connectivity conditions.