Hamilton Circuits in the Directed Butterfly Network

Jean-Claude Bermond, Eric Darrot, Olivier Delmas, Stéphane Perennes* Thème 1 — Réseaux et systèmes Projet SLOOP Rapport de recherche n ̊???? — Juillet 1996 — 25 pages Abstract: in this paper, we prove that the wrapped Butterfly digraph ~ WBF(d; n) of degree d and dimensionn 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 corollary, we have a Hamilton decomposition of ~ WBF(d; n) for any d divisible by a number q, with 4 q 12000. Key-words: Butterfly graph, graph theory, Hamilton decomposition, Hamilton cycle, Hamilton circuit, perfect matching.