Routing permutations and 2–1 routing requests in the hypercube

Abstract

Let H n be the directed symmetric n-dimensional hypercube. Using the computer, we show that for any permutation of the vertices of H 4, there exists a system of pairwise arc-disjoint directed paths from each vertex to its target in the permutation. This verifies Szymanski's conjecture (Proceedings of the International Conference on Parallel Processing, 1989, pp. I-103–I-110) for n=4. We also consider the so-called 2–1 routing requests in H n , 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 H n by arc-disjoint paths: in other words, for n⩾3, H n is not (2–1)-rearrangeable.