A Semiexact Degree Condition for Hamilton Cycles in Digraphs

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We show that for each $\beta > 0$, every digraph G of sufficiently large order n whose outdegree and indegree sequences $d_1^+ \leq \cdots \leq d_n^+$ and $d_1^- \leq \cdots \leq d_n^-$ satisfy $d_i^+, d_i^- \geq \min{\{i + \beta n, n/2\}}$ is Hamiltonian. In fact, we can weaken these assumptions to (i) $d_i^+ \geq \min{\{i + \beta n, n/2\}}$ or $d^-_{n - i - \beta n} \geq n-i$, (ii) $d_i^- \geq \min{\{i + \beta n, n/2\}}$ or $d^+_{n - i - \beta n} \geq n-i$, and still deduce that G is Hamiltonian. This provides an approximate version of a conjecture of Nash-Williams from 1975 and improves a previous result of Kühn, Osthus, and Treglown.