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 Delta is at most n - Delta + 3, and this bound is sharp. We also show that the bound can be improved for Eulerian digraphs with no 2-cycle to n - 2 Delta + O(Delta(2/3)), and we exhibit an infinite family of such digraphs of diameter n - 2 Delta + Omega(Delta(1/2)). We further show that for bipartite Eulerian digraphs with no 2-cycle the diameter is at most n - 2 Delta + 3, and that this bound is sharp. (C) 2015 Elsevier B.V. All rights reserved.