This Article 
   
 Share 
   
 Bibliographic References 
   
 Add to: 
 
Digg
Furl
Spurl
Blink
Simpy
Google
Del.icio.us
Y!MyWeb
 
 Search 
   
k-Pancyclicity of k-ary n-Cube Networks under the Conditional Fault Model
June 2012 (vol. 23 no. 6)
pp. 1115-1120
Jing Li, Taiyuan University of Science and Technology, Taiyuan
Di Liu, Northwestern Polytechnical University, Xi'an City
The k-ary n-cube is one of the most popular interconnection networks for parallel and distributed systems. We prove that a k-ary n-cube with at most 4n-5 faulty edges but where every vertex is incident with at least two healthy edges is k-pancyclic and bipancyclic for n\ge 3 and odd k\ge 3.

[1] Y.A. Ashir and I.A. Stewart, "Fault-Tolerant Embeddings of Hamiltonian Circuits in $k$ -ary $n$ -Cubes," SIAM J. Discrete Math., vol. 15, no. 3, pp. 317-328, 2002.
[2] J.A. Bondy and U.S.R. Murty, Graph Theory. Springer, 2007.
[3] J.S. Fu, "Fault-free Hamiltonian Cycles in Twisted Cubes with Conditional Link Faults," Theoretical Computer Science, vol. 407, nos. 1-3, pp. 318-329, 2008.
[4] J.S. Fu, "Edge-Fault-Tolerant Vertex-Pancyclicity of Augmented Cubes," Information Processing Letters, vol. 110, no. 11, pp. 439-443, 2010.
[5] J.S. Fu, H.S. Hung, and G.H. Chen, "Embedding Fault-Free Cycles in Crossed Cubes with Conditional Link Faults," J. Supercomputing, vol. 49, no. 2, pp. 219-233, 2009.
[6] S.Y. Hsieh and N.W. Chang, "Hamiltonian Path Embedding and Pancyclicity on the Möbius Cube with Faulty Nodes and Faulty Edges," IEEE Trans. Computers, vol. 55, no. 7, pp. 854-863, July 2006.
[7] S.-Y. Hsieh and C.-H. Chen, "Pancyclicity on Mobius Cubes with Maximal Edge Faults," Parallel Computing, vol. 30, no. 3, pp. 407-421, 2004.
[8] S.-Y. Hsieh and Y.-R. Cian, "Conditional Edge-Fault Hamiltonicity of Augmented Cubes," Information Sciences, vol. 180, no. 13, pp. 2596-2617, 2010.
[9] S.-Y. Hsieh, C.-W. Ho, and G.-H. Chen, "Fault-free Hamiltonian Cycles in Faulty Arrangement Graphs," IEEE Trans. Parallel and Distributed Systems, vol. 10, no. 3, pp. 223-237, Mar. 1999.
[10] S.-Y. Hsieh and C.-W. Lee, "Conditional Edge-Fault Hamiltonicity of Matching Composition Networks," IEEE Trans. Parallel and Distributed Systems, vol. 20, no. 4, pp. 581-592, Apr. 2009.
[11] S.Y. Hsieh and C.W. Lee, "Pancyclicity of Restricted Hypercube-Like Networks under the Conditional Fault Model," SIAM J. Discrete Math., vol. 23, no. 4, pp. 2100-2119, 2010.
[12] S.-Y. Hsieh, T.-J. Lin, and H.-L. Huang, "Panconnectivity and Edge-pancyClicity of 3-ary $n$ -Cubes," J. Supercomputing, vol. 42, no. 2, pp. 225-233, 2007.
[13] S.Y. Hsieh and C.D. Wu, "Optimal Fault-Tolerant Hamiltonicity of Star Graphs with Conditional Edge Faults," J. Supercomputing, vol. 49, no. 3, pp. 354-372, 2009.
[14] H.S. Hung, G.H. Chen, and J.S. Fu, "Fault-Free Hamiltonian Cycles in Crossed Cubes with Conditional Link Faults," Information Sciences, vol. 177, no. 24, pp. 5664-5674, 2007.
[15] R.E. Kessler and J.L. Schwarzmeier, "Cray T3D: A New Dimension for Cray Research," Proc. 38th IEEE CS. Int'l Conf. Compcon Spring '93, pp. 176-182, 1993.
[16] C.-N. Kuo and S.-Y. Hsieh, "Pancyclicity and Bipancyclicity of Conditional Faulty Folded Hypercubes," Information Sciences, vol. 180, no. 15, pp. 2904-2914, 2010.
[17] F.T. Leighton, Introduction to Parallel Algorithms and Architectures: Arrays, Trees, Hypercubes. Morgan Kaufmann, 1992.
[18] J. Li, S.Y. Wang, and D. Liu, "Pancyclicity of Ternary N-Cube Networks under the Conditional Fault Model," Information Processing Letters, vol. 111, no. 8, pp. 370-374, 2011.
[19] J. Li, S.Y. Wang, D. Liu, and S.W. Lin, "Edge-Bipancyclicity of the $k$ -ary $n$ -Cubes with Faulty Nodes and Edges," Information Sciences, vol. 181, no. 11, pp. 2260-2267, 2011.
[20] S.W. Lin, S.Y. Wang, and C.F. Li, "Panconnectivity and Edge-Pancyclicity of $k$ -ary $n$ -Cubes with Faulty Elements," Discrete Applied Math., vol. 159, no. 4, pp. 212-223, 2011.
[21] M.J. Ma, G.Z. Liu, and J.M. Xu, "Panconnectivity and Edge-Fault-Tolerant Pancyclicity of Augmented Cubes," Parallel Computing, vol. 33, no. 1, pp. 36-42, 2007.
[22] M. Noakes and W.J. Dally, "System Design of the J-machine," Proc. Sixth MIT Conf. Advanced Research in VLSI, pp. 179-194, 1990.
[23] B. Parhami, An Introduction to Parallel Processing Algorithms and Architectures. Plenum Press, 1999.
[24] J.H. Park, H.S. Lim, and H.C. Kim, "Panconnectivity and Pancyclicity of Hypercube-Like Interconnection Networks with Faulty Elements," Theoretical Computer Science, vol. 377, nos. 1-3, pp. 170-180, 2007.
[25] C. Peterson, J. Sutton, and P. Wiley, "iWarp: A 100-MOPS VLIW Microprocessor for Multicomputers," IEEE Micro, vol. 11, no. 3, pp. 26-37, June 1991.
[26] I.A. Stewart and Y. Xiang, "Embedding Long Paths in $k$ -ary $n$ -Cubes with Faulty Nodes and Links," IEEE Trans. Parallel and Distributed Systems, vol. 19, no. 8, pp. 1071-1085, Aug. 2008.
[27] I.A. Stewart and Y.H. Xiang, "Bipanconnectivity and Bipancyclicity in $k$ -ary $n$ -Cubes." IEEE Trans. Parallel and Distributed Systems, vol. 20, no. 1, pp. 25-33, Jan. 2009.
[28] C.H. Tsai, "Linear Array and Ring Embeddings in Conditional Faulty Hypercubes," Theoretical Computer Science, vol. 314, no. 3, pp. 431-443, 2004.
[29] D. Wang, T. An, M. Pan, K. Wang, and S. Qu, "Hamiltonian-Like Properties of $k$ -ary $n$ -Cubes," Proc. Sixth Int'l Conf. Parallel and Distributed Computing, Applications and Technologies (PDCAT '05), pp. 1002-1007, 2005.
[30] W.W. Wang, M.J. Ma, and J.M. Xu, "Fault-Tolerant Pancyclicity of Augmented Cubes," Information Processing Letters, vol. 103, no. 2, pp. 52-56, 2007.
[31] Y. Xiang and I.A. Stweart, "Pancyclicity in Faulty $k$ -Ary 2-Cubes," Proc. 21th IASTED Int'l Conf. Parallel and Distributed Computing and Systems (PDCS '09), vol. 2-4, pp. 77-84, 2009.
[32] Y.H. Xiang and I.A. Stewart, "Bipancyclicity in $k$ -ary $n$ -Cubes with Faulty Edges under a Conditional Fault Assumption," IEEE Trans. Parallel and Distributed Systems, vol. 22, no. 9, pp. 1506-1513, doi: 10.1109/TPDS.2011.22. Sept. 2011,
[33] M.-C. Yang, J.J.M. Tan, and L.-H. Hsu, "Hamiltonian Circuit and Linear Array Embeddings in Faulty $k$ -ary $n$ -cubes," J. Parallel and Distributed Computing, vol. 67, no. 4, pp. 362-368, 2007.

Index Terms:
Interconnection networks, fault-tolerant, k-ary n-cube, k-pancyclicity, bipancyclicity.
Citation:
Jing Li, Di Liu, "k-Pancyclicity of k-ary n-Cube Networks under the Conditional Fault Model," IEEE Transactions on Parallel and Distributed Systems, vol. 23, no. 6, pp. 1115-1120, June 2012, doi:10.1109/TPDS.2011.211
Usage of this product signifies your acceptance of the Terms of Use.