Hamilton Cycles in Dense Regular Digraphs and Oriented Graphs

Abstract

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 Kühn and Osthus about the Hamiltonicity of regular directed graphs with suitable degree and connectivity conditions.

33 pages, 7 figures