On the maximum number of fault-free mutually independent Hamiltonian cycles in the faulty hypercube

Tzu-Liang Kung (), Cheng-Kuan Lin and Lih-Hsing Hsu
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Tzu-Liang Kung: Asia University
Cheng-Kuan Lin: National Chiao Tung University
Lih-Hsing Hsu: Providence University

Journal of Combinatorial Optimization, 2014, vol. 27, issue 2, No 9, 328-344

Abstract: Abstract Hsieh and Yu (2007) first claimed that an injured n-dimensional hypercube Q n contains (n−1−f)-mutually independent fault-free Hamiltonian cycles, where f≤n−2 denotes the total number of permanent edge-faults in Q n for n≥4, and edge-faults can occur everywhere at random. Later, Kueng et al. (2009a) presented a formal proof to validate Hsieh and Yu’s argument. This paper aims to improve this mentioned result by showing that up to (n−f)-mutually independent fault-free Hamiltonian cycles can be embedded under the same condition. Let F denote the set of f faulty edges. If all faulty edges happen to be incident with an identical vertex s, i.e., the minimum degree of the survival graph Q n −F is equal to n−f, then Q n −F contains at most (n−f)-mutually independent Hamiltonian cycles starting from s. From such a point of view, the presented result is optimal. Thus, not only does our improvement increase the number of mutually independent fault-free Hamiltonian cycles by one, but also the optimality can be achieved.

Keywords: Interconnection network; Graph; Hypercube; Fault tolerance; Hamiltonian cycle (search for similar items in EconPapers)
Date: 2014
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DOI: 10.1007/s10878-012-9528-1

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