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

  • Authors:

  • Tzu-Liang Kung;Cheng-Kuan Lin;Lih-Hsing Hsu

  • Affiliations:

  • Department of Computer Science and Information Engineering, Asia University, Taichung, Taiwan 41354;Department of Computer Science, National Chiao Tung University, Hsinchu, Taiwan 30010;Department of Computer Science and Information Engineering, Providence University, Taichung, Taiwan 43301

  • Venue:
  • Journal of Combinatorial Optimization
  • Year:
  • 2014

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Abstract

Hsieh and Yu (2007) first claimed that an injured n-dimensional hypercube Q n contains (n驴1驴f)-mutually independent fault-free Hamiltonian cycles, where f≤n驴2 denotes the total number of permanent edge-faults in Q n for n驴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驴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 驴F is equal to n驴f, then Q n 驴F contains at most (n驴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.