Hamiltonian degree sequences in digraphs

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