Highly fault-tolerant cycle embeddings of hypercubes |
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Affiliation: | 1. Department of Computer and Information Science, National Chiao Tung University, 1001 Ta Hsueh Road, 300, Hsinchu 30050, Taiwan, ROC;2. Department of Computer Science and Information Engineering, Providence University, 43301, Taiwan, ROC |
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Abstract: | The hypercube Qn 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 l for every even 4 ⩽ l ⩽ 2n 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. |
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