Cycles in highly faulty hypercubes

Abstract

The hypercube Q(n) is one of the most popular networks. In, this paper, we first prove that the n-dimensional hypercube is 2n - 5 conditional fault- bipancyclic. That is, an injured hypercube with up to 2n - 5 faulty links has a cycle of length I for every even 4 <= l <= 2(n) when each node of the hypercube is incident with at least two healthy links. In addition, if a certain node is incident with less than two healthy links, we show that an injured hypercube contains cycles of all even lengths except hamiltonian cycles with up to 2n - 3 faulty links. Furthermore, the above two results are optimal. In conclusion, we find cycles of all possible lengths in injured hypercubes with up to 2n - 5 faulty links under all possible fault distributions.