The Diameter of Almost Eulerian Digraphs

Soares [J. Graph Theory 1992] showed that the well known upper bound 3/delta+1n + O(1) on the diameter of undirected graphs of order n and minimum degree delta also holds for digraphs, provided they are eulerian. In this paper we investigate if similar bounds can be given for digraphs that are, in some sense, close to being eulerian. In particular we show that a directed graph of order n and minimum degree delta whose arc set can be partitioned into s trails, where s <= delta - 2, has diameter at most 3(delta + 1 - s/3)(-1)n + O(1). If s also divides delta - 2, then we show the diameter to be at most 3(delta + 1-(delta-2)s/3(delta-2)+ s)(-1)n + O(1). The latter bound is sharp, apart from an additive constant. As a corollary we obtain the sharp upper bound 3(delta + 1-delta-2/3 delta-5)(-1)n + O(1) on the diameter of digraphs that have an eulerian trail.