Degree sequences forcing Hamilton cycles in directed graphs

We prove the following approximate version of Pósaʹs theorem for directed graphs: every directed graph on n vertices whose in- and outdegree sequences satisfy d i − ⩾ i + o ( n ) and d i + ⩾ i + o ( n ) for all i ⩽ n / 2 has a Hamilton cycle. In fact, we prove that such digraphs are pancyclic (i.e. contain cycles of lengths 2 , … , n ). We also prove an approximate version of Chvátalʹs theorem for digraphs. This asymptotically confirms conjectures of Nash-Williams from 1968 and 1975.