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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
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 555–556
Akl SG (1997) Parallel computation: models and methods. Prentice Hall, NJ
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 〈ftp://ftp.zib.de/pub/zib-publications/reports/TR-96-03.ps〉)
Bermond JC (ed) (1992) Interconnection networks a special issue of discrete applied mathematics, vol 37– 38
Bhuyan L, Agrawal DP (1984) Generalized hypercubes and hyperbus structure for a computer network. IEEE Trans Comput 33:323–333
Bruck J, Cypher R, Soroker D (1994) Embedding cube-connected-cycles graphs into faulty hypercubes. IEEE Trans Comput 43(10):1210–1220
Chan MY, Lee SJ (1991) Distributed fault-tolerant embeddings of rings in hypercubes. J Parallel Distrib Comput 11:63–71
Chan MY, Lee SJ (1993) Fault-tolerant embeddings of complete binary trees in hypercubes. IEEE Trans Parallel Distrib Syst 4(3):540–547
Fu JS, Chen GH (2002) Hamiltonicity of the hierarchical cubic network. Theor Comput Syst 35(1):59–79
Hsieh SY, Chen GH, Ho CW (1999) Fault-free Hamiltonian cycles in faulty arrangement graphs. IEEE Trans Parallel Distrib Syst 10(32):223–237
Hsieh SY, Chen GH, Ho CW (2000) Hamiltonian-laceability of star graphs. Networks 36(4):225–232
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):215–227
Hsieh SY, Chen C-H (2004) Pancyclicity on Möbius cubes with maximal edge faults. Parallel Comput 30(3):407–421
Hsieh SY (2005) Embedding longest fault-free paths onto star graphs with more vertex faults. Theor Comput Sci 337(1–3):370–378
Hsu DF (1993) Interconnection networks and algorithms. A special issue of Networks 23(4)
Jwo JS, Lakshmivarahan S, Dhall SK (1991) Embedding of cycles and grids in star graphs. J Circ, Syst Comput 1(1):43–74
Liaw SC, Chang GJ (1998) Generalized diameters and Rabin numbers of networks. J Comb Optim 2(4):371–384
Leighton FT (1992) Introduction to parallel algorithms and architecture: arrays ⋅ trees ⋅ hypercubes. Morgan Kaufmann, San Mateo, CA
Lewinter M, Widulski W (1997) Hyper-Hamiltonian laceable and caterpillar-spannable product graphs. Comput Math Appl 34(11):99–104
Lin CK, Huang HM, Hsu LH, Bau S (2005) Mutually independent Hamiltonian paths in star networks. Networks 46:110–117
Rowley RA, Bose B (1993) Fault-tolerant ring embedding in deBruijn networks. IEEE Trans Comput 42(12):1480–1486
Simmons G (1978) Almost all n-dimensional rectangular lattices are Hamiltonian laceable. Congressus Numeratium 21:103–108
Sun CM, Lin CK, Huang HM, Hsu LH (2006) Mutually independent Hamiltonian paths and cycles in hypercubes. J Interconnect Netw 7(2):235–255
Tsai CH, Tan JM, Linag T, Hsu LH (2002) Fault-tolerant Hamiltonain laceability of hypercubes. Inf Proc Lett 83:301–306
Tseng YC (1996) Embedding a ring in a hypercube with both faulty links and faulty nodes. Inf Proc Lett 59:217–222
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):1185–1195
Wan PJ (1997) Near-optimal conflict-free channel set assignments for an optical cluster-based hypercube network. J Comb Optim 1(2):179–186
Wang D (2001) Embedding Hamiltonian cycles into folded hypercubes with faulty links. J Parallel Distrib Comput 61(4):545–564
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):1747–1762
Wang NC, Yen CP, Chu CP (2005) Multicast communication in wormhole-routed symmetric networks with Hamiltonian cycle model. J Syst Architec 51(3):165–183
West DB (2001) Introduction to graph theory. Prentice-Hall, Upper Saddle River, NJ 07458
Xu JM (2001) Topological structure and analysis of interconnection networks. Kluwer academic publishers
Xu JM (2004) Wide diameters of Cartesian product graphs and digraphs. J Comb Optim 8(2):171–181
Author information
Authors and Affiliations
Corresponding author
Additional information
An extended abstract of this paper appeared in Proceedings of the 2006 International Conference on Innovative Computing, Information and Control (ICICIC), pp. 288–292, IEEE Computer Society Press.
Rights and permissions
About this article
Cite this article
Hsieh, SY., Yu, PY. Fault-free mutually independent Hamiltonian cycles in hypercubes with faulty edges. J Comb Optim 13, 153–162 (2007). https://doi.org/10.1007/s10878-006-9018-4
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10878-006-9018-4