Hamilton Circuits in the Directed Wrapped Butterfly Network

in this paper, we prove that the wrapped Butterfly digraph ~ WBF(d; n) of degree d and dimension n contains at least d?1 arc-disjoint Hamilton circuits, answering a conjecture of D. Barth. We also conjecture that ~ WBF(d; n) can be decomposed into d Hamilton circuits, except for {d = 2 and n = 2}, {d = 2 and n = 3} and {d = 3 and n = 2}. We show that it suffices to prove the conjecture for d prime and n = 2. Then, we give such a Hamilton decomposition for all primes less than 12000 by a clever computer search, and so, as a corollary, we have a Hamilton decomposition of ~ WBF(d; n) for any d divisible by a number q, with 4 q 12000. Circuits Hamiltoniens dans le réseau Butterfly orienté Résumé : dans cet article, nous prouvons que le graphe Butterfly rebouclé (dans sa version orienté) ~ WBF(d; n) de degré d et de dimension n contient au moins d?1 circuits Hamiltoniens arc-disjoints, répondant a ` une conjecture de D. Barth. Nous conjecturons aussi que ~ WBF(d; n) peut e ˆtre décom-posé en d circuits Hamiltoniens, sauf pour {d = 2 et n = 2}, {d = 2 et n = 3} et {d = 3 et n = 2}. Nous montrons qu'il suffit de prouver cette conjecture pour d premier et n = 2. Puis nous donnons une telle décomposition pour tous les premiers jusqu'a ` 12000 grâce a ` une subtilité de programma-tion, induisant comme corollaire que ~ WBF(d; n) admet une décomposition Hamiltonienne pour tout d divisible par un nombre q tel que 4 q 12000. M any interconnection networks have been proposed as suitable topologies for parallel computers. Among them the Butterfly networks have received particular attention, due to their interesting structure. First, we have to warn the reader that under the name Butterfly and with the same notation, different networks are described. Indeed, while some authors consider the Butterfly networks to be mul-tistage networks used to route permutations, others consider them to be point-to-point networks. In what follows, we will use the term Butterfly for the multistage version and we will use Leighton's terminology [13], namely wrapped Butterfly, for the point-to-point version. Furthermore, these networks can be considered either as undirected or directed. To be complete, we recall that some authors consider only binary Butterfly networks ? the …