All-to-all Routing on Kautz Graphs: Regular Routing Beats Shortest Paths

We study packet routing in the Kautz digraph K(d,D), where every ordered pair of distinct vertices is connected by a unique shortest directed path. The regular routing introduced in earlier work schedules all ordered pairs in tau(d,D) = (D-1)d^(D-2) + D d^(D-1) steps. We show that, for every fixed outdegree d at least 2 and all sufficiently large diameters D, no shortest-path routing scheme can match this makespan. More precisely, we prove that K(d,D) contains an edge whose shortest-path congestion strictly exceeds tau(d,D) when D is sufficiently large. Our construction uses edge-words drawn from a subset of ternary unbordered square-free words, together with a trimming inequality that propagates large congestion at distance D down to shorter distances. Computations for d=2 and small D show that for all D at least 4 there is an edge in K(2,D) with congestion greater than tau(2,D).