Journal of Combinatorial Optimization

, Volume 13, Issue 2, pp 153–162 | Cite as

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

  • Sun-Yuan Hsieh
  • Pei-Yu Yu


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.


Fault-tolerant embedding Graph-theoretic interconnection networks Hamiltonian Hypercubes Mutually independent Hamiltonian cycles 


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  1. Akers SB, Harel D, Krishnamurthy B (1987) The star graph: an attractive alternative to the n-cube. In: Proceedings of international conference on parallel processing St. Charles, IL, pp 555–556Google Scholar
  2. Akl SG (1997) Parallel computation: models and methods. Prentice Hall, NJGoogle Scholar
  3. Ascheuer N (1995) Hamiltonian path problems in the on-line optimization of flexible manufacturing systems. Ph.D. Thesis, University of Technology, Berlin, Germany (also available from 〈〉)Google Scholar
  4. Bermond JC (ed) (1992) Interconnection networks a special issue of discrete applied mathematics, vol 37– 38Google Scholar
  5. Bhuyan L, Agrawal DP (1984) Generalized hypercubes and hyperbus structure for a computer network. IEEE Trans Comput 33:323–333Google Scholar
  6. Bruck J, Cypher R, Soroker D (1994) Embedding cube-connected-cycles graphs into faulty hypercubes. IEEE Trans Comput 43(10):1210–1220CrossRefMathSciNetGoogle Scholar
  7. Chan MY, Lee SJ (1991) Distributed fault-tolerant embeddings of rings in hypercubes. J Parallel Distrib Comput 11:63–71CrossRefGoogle Scholar
  8. Chan MY, Lee SJ (1993) Fault-tolerant embeddings of complete binary trees in hypercubes. IEEE Trans Parallel Distrib Syst 4(3):540–547CrossRefGoogle Scholar
  9. Fu JS, Chen GH (2002) Hamiltonicity of the hierarchical cubic network. Theor Comput Syst 35(1):59–79MathSciNetGoogle Scholar
  10. Hsieh SY, Chen GH, Ho CW (1999) Fault-free Hamiltonian cycles in faulty arrangement graphs. IEEE Trans Parallel Distrib Syst 10(32):223–237CrossRefGoogle Scholar
  11. Hsieh SY, Chen GH, Ho CW (2000) Hamiltonian-laceability of star graphs. Networks 36(4):225–232CrossRefMathSciNetGoogle Scholar
  12. Hsieh SY, Chen G-H, Ho C-W (2001) Longest fault-free paths in star graphs with vertex faults. Theor Comput Sci 262(1–2):215–227CrossRefMathSciNetGoogle Scholar
  13. Hsieh SY, Chen C-H (2004) Pancyclicity on Möbius cubes with maximal edge faults. Parallel Comput 30(3):407–421CrossRefMathSciNetGoogle Scholar
  14. Hsieh SY (2005) Embedding longest fault-free paths onto star graphs with more vertex faults. Theor Comput Sci 337(1–3):370–378CrossRefMathSciNetGoogle Scholar
  15. Hsu DF (1993) Interconnection networks and algorithms. A special issue of Networks 23(4)Google Scholar
  16. Jwo JS, Lakshmivarahan S, Dhall SK (1991) Embedding of cycles and grids in star graphs. J Circ, Syst Comput 1(1):43–74CrossRefGoogle Scholar
  17. Liaw SC, Chang GJ (1998) Generalized diameters and Rabin numbers of networks. J Comb Optim 2(4):371–384CrossRefMathSciNetGoogle Scholar
  18. Leighton FT (1992) Introduction to parallel algorithms and architecture: arrays ⋅ trees ⋅ hypercubes. Morgan Kaufmann, San Mateo, CAGoogle Scholar
  19. Lewinter M, Widulski W (1997) Hyper-Hamiltonian laceable and caterpillar-spannable product graphs. Comput Math Appl 34(11):99–104CrossRefMathSciNetGoogle Scholar
  20. Lin CK, Huang HM, Hsu LH, Bau S (2005) Mutually independent Hamiltonian paths in star networks. Networks 46:110–117MATHCrossRefMathSciNetGoogle Scholar
  21. Rowley RA, Bose B (1993) Fault-tolerant ring embedding in deBruijn networks. IEEE Trans Comput 42(12):1480–1486CrossRefGoogle Scholar
  22. Simmons G (1978) Almost all n-dimensional rectangular lattices are Hamiltonian laceable. Congressus Numeratium 21:103–108MathSciNetGoogle Scholar
  23. Sun CM, Lin CK, Huang HM, Hsu LH (2006) Mutually independent Hamiltonian paths and cycles in hypercubes. J Interconnect Netw 7(2):235–255CrossRefGoogle Scholar
  24. Tsai CH, Tan JM, Linag T, Hsu LH (2002) Fault-tolerant Hamiltonain laceability of hypercubes. Inf Proc Lett 83:301–306CrossRefGoogle Scholar
  25. Tseng YC (1996) Embedding a ring in a hypercube with both faulty links and faulty nodes. Inf Proc Lett 59:217–222CrossRefMathSciNetGoogle Scholar
  26. Tseng YC, Chang SH, Sheu JP (1997) Fault-tolerant ring embedding in star graphs with both link and node failures. IEEE Trans Parallel Distrib Syst 8(12):1185–1195CrossRefGoogle Scholar
  27. Wan PJ (1997) Near-optimal conflict-free channel set assignments for an optical cluster-based hypercube network. J Comb Optim 1(2):179–186CrossRefMathSciNetGoogle Scholar
  28. Wang D (2001) Embedding Hamiltonian cycles into folded hypercubes with faulty links. J Parallel Distrib Comput 61(4):545–564CrossRefGoogle Scholar
  29. Wang NC, Chu CP, Chen TS (2002) A dual-Hamiltonian-path-based multicasting strategy for wormhole-routed star graph interconnection networks. J Parallel Distrib Comput 62(12):1747–1762CrossRefGoogle Scholar
  30. Wang NC, Yen CP, Chu CP (2005) Multicast communication in wormhole-routed symmetric networks with Hamiltonian cycle model. J Syst Architec 51(3):165–183CrossRefGoogle Scholar
  31. West DB (2001) Introduction to graph theory. Prentice-Hall, Upper Saddle River, NJ 07458Google Scholar
  32. Xu JM (2001) Topological structure and analysis of interconnection networks. Kluwer academic publishersGoogle Scholar
  33. Xu JM (2004) Wide diameters of Cartesian product graphs and digraphs. J Comb Optim 8(2):171–181CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2006

Authors and Affiliations

  1. 1.Department of Computer Science and Information EngineeringNational Cheng Kung UniversityTainanTaiwan

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