4-cycles in mixing digraphs

It is known that every simple graph with n 3 / 2 edges contains a 4-cycle. A similar statement for digraphs is not possible since no condition on the number of arcs can guarantee an (oriented) 4-cycle. We find a condition which does guarantee the presence of a 4-cycle and our result is tight. Our condition, which we call f-mixing, can be seen as a quasirandomness condition on the orientation of the digraph. We also investigate the notion of mixing for regular and almost regular digraphs. In particular we determine how mixing a random orientation of a random graph is.