On Cycle Embeddings of Cube Families

In this dissertation, we investigate fault-tolerant capabilities of the k-ary n-cubes with respect to the hamiltonian and hamiltonian connected properties. The k-ary n-cube is a bipartite graph if and only if k is an even integer. Let F be a faulty set with nodes and/or links, and let k ≥ 3 be an odd integer. When |F | ≤ 2n − 2, we show that there exists a hamiltonian cycle in a wounded k-ary n-cube. In addition, when |F | ≤ 2n − 3, we prove that, for two arbitrary nodes, there exists a hamiltonian path connecting these two nodes in a wounded k-ary n-cube. Since the k-ary n-cube is regular of degree 2n, the degrees of fault-tolerance 2n − 3 and 2n − 2 respectively, are optimal in the worst case. The Möbius cube MQn, crossed cube CQn, and twisted cube TQn are alternatives to the popular hypercube network. However, the diameters of these three interconnection networks are about one half of that of the hypercube. Recently, MQn was shown to be pancyclic, i.e., cycles of any length at least four can be embedded into it. Due to the importance of the fault tolerance in the parallel processing area, in this dissertation, we study injured MQn, CQn, and TQn with mixed node and link faults. We show that they are (n − 2)-fault-tolerant pancyclic for n ≥ 3, that is, injured n-dimensional MQn, CQn, and TQn are still pancyclic with up to (n−2) faults. Furthermore, our results are optimal in the sense that if there are n − 1 faults, there is no guarantee of having a cycle of a certain length in them. The hypercube Qn is one of the most popular networks. In this dissertation , 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 l for every even 4 ≤ l ≤ 2 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.