  This paper concerns all-to-all network routing on regular digraphs. In
previous work we focused on efficient routing in highly symmetric digraphs with
low diameter for fixed degree. Here, we show that every connected regular
digraph has an all-to-all routing scheme and associated schedule with no
waiting. In fact, this routing scheme becomes more efficient as the diameter
goes down with respect to the degree and number of vertices. Lastly, we examine
the simple scheduling algorithm called ``farthest-distance-first'' and prove
that it yields optimal schedules for all-to-all communication in networks of
interest, including Kautz graphs.
