The foundation of information society is computer interconnection network, and the key of information exchange is communication algorithm. Finding interconnection networks with simple routing algorithm and high fault-tolerant performance is the premise of realizing various communication algorithms and protocols. Nowadays, people can build complex interconnection networks by using very large scale integration (VLSI) technology. Locally exchanged twisted cubes, denoted by (s + t + 1)-dimensional LeTQs,t, which combines the merits of the exchanged hypercube and the locally twisted cube. It has been proved that the LeTQs,t has many excellent properties for interconnection networks, such as fewer edges, lower overhead and smaller diameter. Embeddability is an important indicator to measure the performance of interconnection networks. We mainly study the fault tolerant Hamiltonian properties of a faulty locally exchanged twisted cube, LeTQs,t − (fv + fe), with faulty vertices fv and faulty edges fe. Firstly, we prove that an LeTQs,t can tolerate up to s − 1 faulty vertices and edges when embedding a Hamiltonian cycle, for s ⩾ 2, t ⩾ 3, and s ⩽ t. Furthermore, we also prove another result that there is a Hamiltonian path between any two distinct fault-free vertices in a faulty LeTQs,t with up to (s − 2) faulty vertices and edges. That is, we show that LeTQs,t is (s − 1)-Hamiltonian and (s − 2)-Hamiltonian-connected. The results are proved to be optimal in this paper with at most (s − 1)-fault-tolerant Hamiltonicity and (s − 2) fault-tolerant Hamiltonian connectivity of LeTQs,t.
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Lin L M, Hsieh S Y, Xu L, Zhou S M, Chen R Q. The relationship between extra connectivity and conditional diagnosability of regular graphs under the PMC model. Journal of Computer System and Sciences, 2018, 95: 1–18
Guo L. Reliability analysis of twisted cubes. Theoretical Computer Science, 2018, 707: 96–101
Lin L M, Hsieh S Y, Chen R Q, Xu L, Lee C W. The relationship between g-restricted connectivity and g-good-neighbor fault diagnosability of general regular networks. IEEE Transactions on Reliability, 2018, 67(1): 285–296
Wang D. Hamiltonian embedding in crossed cubes with failed links. IEEE Transactions on Parallel and Distributed Systems, 2012, 23(11): 2117–2124
Fan J, Jia X, Lin X. Embedding of cycles in twisted cubes with edge-pancyclic. Algorithmica, 2008, 51(3): 264–282
Liu Z, Fan J, Zhou J, Cheng B, Jia X. Fault-tolerant embedding of complete binary trees in locally twisted cubes. Journal of Parallel and Distributed Computing, 2017, 101: 69–78
Wei C C, Hsieh S Y. h-Restricted connectivity of locally twisted cubes. Discrete Applied Mathematics, 2017, 217: 330–339
Huang Y, Lin L, Wang D, Xu L. Minimum neighborhood of alternating group graphs. IEEE Access, 2019, 7: 17299–17311
Zhai Y, Lin L, Xu L, Zhang X, Huang Y. The conditional diagnosability with g-good-neighbor of exchanged hypercubes. The Computer Journal, 2019, 62(5): 747–756
Zhou D F, Fan J X, Lin C K, Cheng B L, Zhou J Y, Liu Z. Optimal path embedding in the exchanged crossed cube. Journal of Computer Science and Technology, 2017, 32 (3): 618–629
Ren Y, Wang S. The g-good-neighbour diagnosability of locally twisted cubes. Theoretical Computer Science, 2017, 697: 91–97
Yang X, Evans D J, Megson G M. The locally twisted cubes. International Journal of Computer Mathematics, 2005, 82(4): 401–413
Fan W B, Fan J X, Lin C K, Wang G J, Cheng B L, Wang R C. An efficient algorithm for embedding exchanged hypercubes into grids. The Journal of Supercomputing, 2019, 75(2): 783–807
Loh P K K, Hsu W J, Pan Y. The exchanged hypercube. IEEE Transactions on Parallel and Distributed Systems, 2005, 16(9): 866–874
Zhang Z, Deng Y, Min G, Xie J, Huang S. ExCCC-DCN: a highly scalable, cost-effective and energy-efficient data center structure. IEEE Transactions on Parallel and Distributed Systems, 2017, 28(4): 1046–1060
Chang J M, Chen X R, Yang J S, Wu R Y. Locally exchanged twisted cubes: connectivity and super connectivity. Information Processing Letters, 2016, 116(7): 460–466
Fan W B, Fan J X, Lin C K, Wang Y, Han Y J, Wang R C. Optimally embedding 3-ary n-cubes into grids. Journal of Computer Science and Technology, 2019, 34(2): 372–387
Lin L M, Xu L, Zhou S M, Hsieh S Y. The t/k-Diagnosability for regular networks. IEEE Transactions on Computers, 2016, 65(10): 3157–3170
Lin L M, Xu L, Zhou S M, Hsieh S Y. The extra, restricted connectivity and conditional diagnosability of split-star networks. IEEE Transactions on Parallel and Distributed Systems, 2016, 27(2): 533–545
Cheng E, Qiu K, Shen S. Diagnosability problems of the exchanged hypercube and its generalization. International Journal of Computer Mathematics: Computer Systems Theory, 2017, 2(2): 39–52
Cheng E, Qiu K, Shen Z. A strong connectivity property of the generalized exchanged hypercube. Discrete Applied Mathematics, 2017, 216: 529–536
Guo L, Su G, Lin W, Chen J. Fault tolerance of locally twisted cubes. Applied Mathematics and Computation, 2018, 334: 401–406
Guo L, Guo X. Fault tolerance of hypercubes and folded hypercubes. The Journal of Supercomputing, 2014, 68(3): 1235–1240
Wei C C, Chen C A, Hsieh S Y. Conditional (t, k)-diagnosis in regular and irregular graphs under the comparison diagnosis model. IEEE Transactions on Dependable Security and Computation, 2018, 15(2): 351–356
Lin L M, Xu L, Zhou S M, Xiang Y, Trustworthiness-hypercube-based reliable communication in mobile social networks. Information Sciences, 2016, 369: 34–50
Huang Y, Lin L, Wang D. On the reliability of alternating group graph-based networks. Theoretical Computer Science, 2018, 728: 9–28
Li X, Liu B, Ma M, Xu J. Many-to-many disjoint paths in hypercubes with faulty vertices. Discrete Applied Mathematics, 2017, 217: 229–242
Lu H, Wang F. Hamiltonian paths passing through prescribed edges in balanced hypercubes. Theoretical Computer Science, 2019, 761: 23–33
Liu H, Hu X, Gao S. Hamiltonian cycles and paths in faulty twisted hypercubes. Discrete Applied Mathematics, 2019, 257: 243–249
Xu X, Huang Y, Zhang P, Zhang S. Fault-tolerant vertex-pancyclicity of locally twisted cubes LTQn. Journal of Parallel and Distributed Computing, 2016, 88: 57–62
Cheng D, Hao R. Various cycles embedding in faulty balanced hypercubes. Information Sciences, 2015, 297: 140–153
Cheng C W, Hsieh S Y. Fault-tolerant cycle embedding in cartesian product graphs: edge-pancyclicity and edge-bipancyclicity with faulty edges. IEEE Transactions on Parallel and Distributed Systems, 2015, 26(11): 2997–3011
Lv Y L, Lin C K, Fan J X, Jia X H. Hamiltonian cycle and path embeddings in 3-ary n-cubes based on K1,3-structure faults. Journal of Parallel and Distributed Computing, 2018, 120: 148–158
Garey M R, Johnson D S. Computers and Intractability: a Guide to The Theory of NP-Completeness. United States of America: W. H. Freeman and Company, 1979
Hsieh S Y, Wu C Y. Edge-fault-tolerant hamiltonicity of locally twisted cubes under conditional edge faults. Journal of Combinatorial Optimization, 2010, 19(1): 16–30
Xu X, Zhai W, Xu J, Deng A, Yang Y. Fault-tolerant edge-pancyclicity of locally twisted cubes. Information Sciences, 2011, 181(11): 2268–2277
We would like to express our sincerest appreciation to Prof. Guoliang Chen for his constructive suggestions. This work was supported by the National Natural Science Foundation of China (Grant Nos. U1905211, 61872196, 61902195 and 61972272), Natural Science Foundation of Jiangsu Province (BK20200753), Natural Science Fund for Colleges and Universities in Jiangsu Province (General Program, 19KJB520045), NUPTSF (NY219151, NY219131).
Weibei Fan received the BS and MS degrees from Henan University of Urban Construction and Henan University, China in 2011 and 2014, respectively, and received the PhD degree in Computer Science from Soochow University, China from 2019. He is currently an associate professor incomputerscience at Nanjing University of Posts and Telecommunications, China. His research interests include parallel and distributed systems, cloud computing, and interconnection networks.
Jianxi Fan received the BS, MS, and PhD degrees in Computer Science from the Shandong Normal University, Shandong University, and the City University of Hong Kong, China in 1988, 1991, and 2006, respectively. He is currently a professor in the School of Computer Science and Technology at the Soochow University, China. He was a visiting scholar in the Department of Computer Science at Montclair State University (May 2017–August 2017) and a senior research fellow in the Department of Computer Science at the City University of Hong Kong (May 2012–August 2012). His research interests include parallel and distributed systems, interconnection architectures, data center networks, algorithms, and graph theory.
Zhijie Han received the MS and PhD degrees computer science from Henan University and Soochow University, China in 2006 and 2009, respectively. He is currently a vice professor in the school of computer and information engineering, Henan University, China. His research interests include parallel and distributed computing, cloud computing, and big data.
Peng Li received the PhD degree from Nanjing University of Posts and Telecommunications (NUPT), China in 2013. Currently, he is an associate professor and Master Supervisor in NUPT. He has managed a number of research projects supported by the National Natural Science Foundation of China and research projects supported by the Ministry of Jiangsu Province. His main research interests include parallel and distributed computing, cloud computing computer communication networks (especially in Wireless Sensor Networks, Ad Hoc Network) and information security.
Yujie Zhang received the BS and MS degrees from Henan Polytechnic University and Henan University, China in 2012 and 2016, respectively. He is currently working toward the PhD degree in information security at Nanjing University of Posts and Telecommunications, China. His research interests include cloud computing, cooperative communications, and information security.
Ruchuan Wang researched on graphic processing at the University of Bremen and program design theory Ludwig Maximilian Muenchen Unitversitaet, German from 1984 to 1992. He is currently a professor and the PhD supervisor in the School of Computer, Nanjing University of Posts and Telecommunications, China. His major research interests include parallel and distributed systems, wireless sensor networks and information security.
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Fan, W., Fan, J., Han, Z. et al. Fault-tolerant hamiltonian cycles and paths embedding into locally exchanged twisted cubes. Front. Comput. Sci. 15, 153104 (2021). https://doi.org/10.1007/s11704-020-9387-3
- interconnection network
- LeTQ s,t
- hamiltonian cycle
- hamiltonian path