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.

中文翻译：

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稠密正则有向图和有向图中的哈密顿循环

我们证明对于每一个$\epsilon >0$那里存在$n_0=n_0（\epsilon ）$这样每个正则导向图$n>n_0$至少有顶点和度数$（1/4+\epsilon ）n$有汉密尔顿循环。这建立了 Jackson 1981 年猜想的近似版本。我们还建立了与 Kühn 和 Osthus 关于具有适当度和连通性条件的正则有向图的哈密顿性的猜想相关的结果。