Hamiltonian degree sequences in digraphs
  We show that for each \eta>0 every digraph G of sufficiently large order n is Hamiltonian if its out- and indegree sequences d^+_1\le ... \le d^+_n and d^- _1 \le ... \le d^-_n satisfy   (i) d^+_i \geq i+ \eta n or d^-_{n-i- \eta n} \geq n-i and   (ii) d^-_i \geq i+ \eta n or d^+_{n-i- \eta n} \geq n-i for all i < n/2.   This gives an approximate solution to a problem of Nash-Williams concerning a digraph analogue of Chv\'atal's theorem. In fact, we prove the stronger result that such digraphs G are pancyclic. 
Daniela K\"uhn, Deryk Osthus and Andrew Treglown