Hamilton cycles in dense regular digraphs and oriented graphs

We prove that for every epsilon > 0 there exists n(0) = n(0)(epsilon) such that every regular oriented graph on n > n(0) vertices and degree at least (1/4 + epsilon)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 Kuhn and Osthus about the Hamiltonicity of regular directed graphs with suitable degree and connectivity conditions.(c) 2023 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons .org /licenses /by /4 .0/).