2-1 routing requests in the hypercube

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Abstract

Let Hn be the directed symmetric n-dimensional hypercube. We consider in Hn the so-called 2-1 routing requests, where any vertex of Hn can be used twice as a source, but only once as a target. In order to disprove the Szymanski's conjecture for a given dimension n, it is necessary to obtain 2-1 routing requests in Hn1 that cannot be routed.

In [O. Baudon, G. Fertin, and I. Havel, Routing permutations in the hypercube, Discrete Applied Mathematics 113 (1) (September 2001) 43–58], we showed that in H3 there exists exactly two routing requests which cannot be routed, nonequivalent by automorphism. Moreover, we showed that for one of them, called g3, it is possible to extend it for any dimension n3 in a 2-1 routing request gn that cannot be routed in Hn.

Considering distances between sources and their respective target in g4, we have, using a computer, studying all the 2-1 routing requests of H4 having similar properties on distances, in order to find more not routable 2-1 routing request in dimension 4 and higher. We have by this way obtained several (a dozen) non routable 2-1 rout- ing requests in H4. Some of them may be extended to higher dimensions, but not all.

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