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 2 δ + 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 2 δ + 1 n - 4 . We show that Knyazevʹs upper bound can be improved to diam ( D ) ⩽ 4 2 δ + 1 n + 2 , and this bound is sharp apart from an additive constant.