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

Author

Listed:
  • Tzu-Liang Kung

    (Asia University)

  • Cheng-Kuan Lin

    (National Chiao Tung University)

  • Lih-Hsing Hsu

    (Providence University)

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.

Suggested Citation

  • Tzu-Liang Kung & Cheng-Kuan Lin & Lih-Hsing Hsu, 2014. "On the maximum number of fault-free mutually independent Hamiltonian cycles in the faulty hypercube," Journal of Combinatorial Optimization, Springer, vol. 27(2), pages 328-344, February.
    • Handle: RePEc:spr:jcomop:v:27:y:2014:i:2:d:10.1007_s10878-012-9528-1
    DOI: 10.1007/s10878-012-9528-1
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    References listed on IDEAS

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    1. Sun-Yuan Hsieh & Pei-Yu Yu, 2007. "Fault-free mutually independent Hamiltonian cycles in hypercubes with faulty edges," Journal of Combinatorial Optimization, Springer, vol. 13(2), pages 153-162, February.
    2. Nelson Castañeda & Ivan S. Gotchev, 2010. "Embedded paths and cycles in faulty hypercubes," Journal of Combinatorial Optimization, Springer, vol. 20(3), pages 224-248, October.
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    Cited by:

    1. Yingbo Wu & Tianrui Zhang & Shan Chen & Tianhui Wang, 2017. "The Minimum Spectral Radius of an Edge-Removed Network: A Hypercube Perspective," Discrete Dynamics in Nature and Society, Hindawi, vol. 2017, pages 1-8, April.

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