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Fault-free mutually independent Hamiltonian cycles in hypercubes with faulty edges

Author

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  • Sun-Yuan Hsieh

    (National Cheng Kung University)

  • Pei-Yu Yu

    (National Cheng Kung University)

Abstract

Two Hamiltonian paths are said to be fully independent if the ith vertices of both paths are distinct for all i between 1 and n, where n is the number of vertices of the given graph. Hamiltonian paths in a set are said to be mutually fully independent if two arbitrary Hamiltonian paths in the set are fully independent. On the other hand, two Hamiltonian cycles are independent starting at v if both cycles start at a common vertex v and the ith vertices of both cycles are distinct for all i between 2 and n. Hamiltonian cycles in a set are said to be mutually independent starting at v if any two different cycles in the set are independent starting at v. The n-dimensional hypercube is widely used as the architecture for parallel machines. In this paper, we study its fault-tolerant property and show that an n-dimensional hypercube with at most n−2 faulty edges can embed a set of fault-free mutually fully independent Hamiltonian paths between two adjacent vertices, and can embed a set of fault-free mutually independent Hamiltonian cycles starting at a given vertex. The number of tolerable faulty edges is optimal with respect to a worst case.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jcomop:v:13:y:2007:i:2:d:10.1007_s10878-006-9018-4
    DOI: 10.1007/s10878-006-9018-4
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    References listed on IDEAS

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    1. Sheng-Chyang Liaw & Gerard J. Chang, 1998. "Generalized Diameters and Rabin Numbers of Networks," Journal of Combinatorial Optimization, Springer, vol. 2(4), pages 371-384, December.
    2. Jun-Ming Xu, 2004. "Wide Diameters of Cartesian Product Graphs and Digraphs," Journal of Combinatorial Optimization, Springer, vol. 8(2), pages 171-181, June.
    3. Peng-Jun Wan, 1997. "Near-Optimal Conflict-Free Channel Set Assignments for an Optical Cluster-Based Hypercube Network," Journal of Combinatorial Optimization, Springer, vol. 1(2), pages 179-186, June.
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    Cited by:

    1. Gregor, Petr & Škrekovski, Riste & Vukašinović, Vida, 2015. "Rooted level-disjoint partitions of Cartesian products," Applied Mathematics and Computation, Elsevier, vol. 266(C), pages 244-258.
    2. 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.
    3. Gregor, Petr & Škrekovski, Riste & Vukašinović, Vida, 2018. "Modelling simultaneous broadcasting by level-disjoint partitions," Applied Mathematics and Computation, Elsevier, vol. 325(C), pages 15-23.
    4. Tz-Liang Kueng & Cheng-Kuan Lin & Tyne Liang & Jimmy J. M. Tan & Lih-Hsing Hsu, 2009. "A note on fault-free mutually independent Hamiltonian cycles in hypercubes with faulty edges," Journal of Combinatorial Optimization, Springer, vol. 17(3), pages 312-322, April.

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