A semi-exact degree condition for Hamilton cycles in digraphs

The paper is concerned with directed versions of Posa's theorem and Chvatal's theorem on Hamilton cycles in graphs. We show that for each a>0, every digraph G of sufficiently large order n whose outdegree and indegree sequences d_1^+ \leq ... \leq d_n^+ and d_1^- \leq >... \leq d_n^- satisfy d_i^+, d_i^- \geq min{i + a n, n/2} is Hamiltonian. In fact, we can weaken these assumptions to (i) d_i^+ \geq min{i + a n, n/2} or d^-_{n - i - a n} \geq n-i; (ii) d_i^- \geq min{i + a n, n/2} or d^+_{n - i - a 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 K\"uhn, Osthus and Treglown.