Hamiltonian degree sequences in digraphs

We show that for each η > 0 every digraph G of sufficiently large order n is Hamiltonian if its out- and indegree sequences d 1 + ⩽ ⋯ ⩽ d n + and d 1 − ⩽ ⋯ ⩽ d n − satisfy (i) d i + ⩾ i + η n or d n − i − η n − ⩾ n − i and (ii) d i − ⩾ i + η n or d n − i − η n + ⩾ n − i for all i < n / 2 . This gives an approximate solution to a problem of Nash-Williams (1975) [22] concerning a digraph analogue of Chvátalʹs theorem. In fact, we prove the stronger result that such digraphs G are pancyclic.