Diameter and maximum degree in Eulerian digraphs

In this paper we consider upper bounds on the diameter of Eulerian digraphs containing a vertex of large degree. Define the maximum degree of a digraph to be the maximum of all in-degrees and out-degrees of its vertices. We show that the diameter of an Eulerian digraph of order n and maximum degree Δ is at most n-Δ+3, and this bound is sharp. We also show that the bound can be improved for Eulerian digraphs with no 2-cycle to n-2Δ+O(Δ^2/3), and we exhibit an infinite family of such digraphs of diameter n-2Δ+Ω(Δ^1/2). We further show that for bipartite Eulerian digraphs with no 2-cycle the diameter is at most n-2Δ+3, and that this bound is sharp.