Routing permutations and 2-1 routing requests in the hypercube
Résumé
Let Hn be the directed symmetric n-dimensional hypercube. Using the computer, we show that for any permutation of the vertices of H4, there exists a system of pairwise arc-disjoint directed paths from each vertex to its target in the permutation. This verifies Szymanski's conjecture for n=4. We also consider the so-called 2-1 routing requests in Hn, where any vertex can be used twice as a source but only once as a target; we construct for any n≥3 a 2-1 request that cannot be routed in Hn by arc-disjoint paths:in other words, for n≥3, Hn is not (2-1)-rearrangeable.