The diameter of directed graphs

Let D be a strongly connected oriented graph, i.e., a digraph with no cycles of length 2, of order n and minimum out-degree δ. Let D be eulerian, i.e., the in-degree and out-degree of each vertex are equal. Knyazev (Mat. Z. 41(6) 1987 829) proved that the diameter of D is at most 5/2δ+2 n and, for given n and δ, constructed strongly connected oriented graphs of order n which are δ-regular and have diameter greater than 4/2δ+1 n - 4. We show that Knyazev's upper bound can be improved to diam(D) ≤ 4/2δ+1 n + 2, and this bound is sharp apart from an additive constant.