Abstract
The locally twisted cube is a variation of hypercube, which possesses some properties superior to the hypercube. In this paper, we investigate the edge-fault-tolerant hamiltonicity of an n-dimensional locally twisted cube, denoted by LTQ n . We show that for any LTQ n (n≥3) with at most 2n−5 faulty edges in which each node is incident to at least two fault-free edges, there exists a fault-free Hamiltonian cycle. We also demonstrate that our result is optimal with respect to the number of faulty edges tolerated.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Akl SG (1997) Parallel computation: models and methods. Prentice Hall, New York
Ascheuer N (1995) Hamiltonian path problems in the on-line optimization of flexible manufacturing systems. PhD Thesis, University of Technology, Berlin, Germany (also available from ftp://ftp.zib.de/pub/zib-publications/reports/TR-96-03.ps)
Cull P, Larson SM (1995) The Möbius cubes. IEEE Trans Comput 44(5):647–659
Day K, Al-Ayyoub AE (1997) Fault diameter of k-ary n-cube networks. IEEE Trans Parallel Distrib Syst 8(9):903–907
Diks K, Pele A (1996) Efficient gossiping by packets in networks with random faults. SIAM J Discrete Math 9(1):7–18
Efe K (1992) The crossed cube architecture for parallel computation. IEEE Trans Parallel Distrib Syst 3(5):513–524
Esfahanian AH, Hakimi SL (1958) Fault-tolerant routing in de Bruijn communication networks. IEEE Trans Comput C-34(9):777–788
Fu J-S (2003) Fault-tolerant cycle embedding in the hypercube. Parallel Comput 29(6):821–832
Fu J-S (2006) Conditional fault-tolerant Hamiltonicity of twisted cubes. In: Proceedings of 7th parallel and distributed computing, applications and technologies (PDCAT’06). IEEE Comput Soc, Los Alamitos, pp 5–10
Fu J-S (2007) Conditional fault-tolerant Hamiltonicity of star graphs. Parallel Comput 33:488–496
Hilbers PAJ, Koopman MRJ, van de Snepscheut JLA (1987) The twisted cube. In: Proceedings of the conference on parallel architectures and languages Europe, vol I: Parallel architectures. Lecture notes in computer science, vol 258, pp 152–159
Hillis DW (1982) The connection machine. MIT Press, Cambridge
Hsieh SY (2005) Embedding longest fault-free paths onto star graphs with more vertex faults. Theor Comput Sci 337(1–3):370–378
Hsieh SY (2006) Fault-Tolerant cycle embedding in the hypercube with more both faulty vertices and faulty edges. Parallel Comput 32(1):84–91
Hsieh SY, Wu CD (2007) Conditional edge-fault-tolerant Hamiltonian cycle embedding of star graphs. In: Proceedings of the 13th international conference on parallel and distributed systems (ICPADS’07). IEEE Comput Soc, Los Alamitos
Hsieh SY, Ho CW, Chen GH (1999) Fault-free Hamiltonian cycles in faulty arrangement graphs. IEEE Trans Parallel Distrib Syst 10(3):223–237
Hsieh SY, Chen GH, Ho CW (2001a) Longest fault-free paths in star graphs with vertex faults. Theor Comput Sci 262(1–2):215–227
Hsieh SY, Chen GH, Ho CW (2001b) Longest fault-free paths in star graphs with edge faults. IEEE Trans Comput 50(9):960–971
Hung HS, Chen GH, Fu JS (2007) Fault-free Hamiltonian cycles in crossed cubes with conditional link faults. Inf Sci 177(24):5664–5674
Leighton FT (1992) Introduction to parallel algorithms and architecture: arrays, trees, hypercubes. Morgan Kaufmann, San Mateo
Liang AC, Bhattacharya S, Tsai WT (1994) Fault-tolerant multicasting on hypercubes. J Parallel Distrib Comput 23:418–428
Parhami B (1999) Introduction to parallel processing: algorithms and architectures. Plenum, New York
Park JH, Kim HC, Lim HS (2005) Fault-Hamiltonicity of hypercube-like interconnection networks. In: Proceedings of the 19th IEEE International parallel and distributed processing symposium, 2005 (IPDPS’05), vol 1, pp 60.1
Singhvi N, Ghose K (1995) The Mcube: a symmetrical cube based network with twisted links. In: Proceedings of the 9th international symposium of parallel processing, Honolulu, pp 11–16
Tsai CH (2004) Linear array and ring embeddings in conditional faulty hypercubes. Theor Comput Sci 314:431–443
Tseng YC, Chang SH, Sheu JP (1997) Fault-tolerant ring embedding in a star graph with both link and node failures. IEEE Trans Parallel Distrib Syst 8(12):1185–1195
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 Archit 51(3):165–183
Wu J (1995) Safety levels—an efficient mechanism for achieving reliable broadcasting in hypercubes. IEEE Trans Comput 44(5):702–706
Yang PJ, Tien SB, Raghavendra CS (1994) Embedding of rings and meshes onto faulty hypercubes using free dimensions. IEEE Trans Comput 43(5):608–613
Yang XF, Evans DJ, Megson GM (2005) The locally twisted cubes. Int J Comput Math 82(4):401–413
Author information
Authors and Affiliations
Corresponding author
Additional information
An extended abstract of this paper under the title “Fault-Free Hamiltonian Cycles in Locally Twisted Cubes under Conditional Edge Faults” appeared in Proceedings of the 13th International Conference on Parallel and Distributed Systems (ICPADS), 2007.
Rights and permissions
About this article
Cite this article
Hsieh, SY., Wu, CY. Edge-fault-tolerant hamiltonicity of locally twisted cubes under conditional edge faults. J Comb Optim 19, 16–30 (2010). https://doi.org/10.1007/s10878-008-9157-x
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10878-008-9157-x