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One-to-one disjoint path covers in hypercubes with faulty edges

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Abstract

A one-to-one k-disjoint path cover \(\{P_1,P_2,\ldots ,P_k\}\) of a graph G is a collection of k internally vertex disjoint paths joining source with sink that cover all vertices of G. In this paper, we investigate the problem of one-to-one disjoint path cover in hypercubes with faulty edges and obtain the following results: Let u, v ∈ V(Qn) be such that \(p(u)\ne p(v)\) and \(1\le k\le n\). Then there exists a one-to-one k-disjoint path cover \(\{P_1,P_2,\ldots ,P_k\}\) joining vertices u and v in \(Q_n\). Moreover, when \(1\le k\le n-2\), the result still holds even if removing \(n-2-k\) edges from \(Q_n\).

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References

  1. Arabnia HR (1990) A parallel algorithm for the arbitrary rotation of digitized images using process-and-data-decomposition approach. J Parallel Distrib Comput 10(2):188–193

    Article  Google Scholar 

  2. Arabnia HR, Bhandarkar SM (1996) Parallel stereocorrelation on a reconfigurable multi-ring network. J Supercomput 10(3):243–270 (Special Issue on Parallel and Distributed Processing)

    Article  MATH  Google Scholar 

  3. Arabnia HR, Taha TR (1998) A parallel numerical algorithm on a reconfigurable multi-ring network. J Telecommun Syst 10:185–203 (Special issue on Interconnections Networks)

    Article  Google Scholar 

  4. Bhandarkar SM, Arabnia HR (1995) The Hough transform on a reconfigurable multi-ring network. J Parallel Distrib Comput 24(1):107–114

    Article  Google Scholar 

  5. Bhandarkar SM, Arabnia HR (1995) The REFINE multiprocessor theoretical properties and algorithms. Parallel Comput 21(11):1783–1806

    Article  Google Scholar 

  6. Bhandarkar SM, Arabnia HR (1997) Parallel computer vision on a reconfigurable multiprocessor network. IEEE Trans Parallel Distrib Syst 8(3):292–310

    Article  Google Scholar 

  7. Bondy J, Murty U (1976) Graph theory with applications. Macmillan Press, London

    Book  MATH  Google Scholar 

  8. Caha R, Koubek V (2007) Spanning multi-paths in hypercubes. Discret Math 307:2053–2066

    Article  MathSciNet  MATH  Google Scholar 

  9. Cao H, Zhang B, Zhou Z (2018) One-to-one disjoint path covers in digraphs. Theor Comput Sci 714:27–35

    Article  MathSciNet  MATH  Google Scholar 

  10. Castaneda N, Gotchev I (2010) Embedded paths and cycles in faulty hypercubes. J Combin Optim 20(3):224–248

    Article  MathSciNet  MATH  Google Scholar 

  11. Chen XB (2012) Paired many-to-many disjoint path covers of hypercubes with faulty edges. Inf Process Lett 112:61–66

    Article  MathSciNet  MATH  Google Scholar 

  12. Chen XB (2013) Paired many-to-many disjoint path covers of the hypercubes. Inf Sci 236(1):218–223

    Article  MathSciNet  MATH  Google Scholar 

  13. Cheng D, Hao R, Feng Y (2014) Two node-disjoint paths in balanced hypercubes. Appl Math Comput 242:127–142

    MathSciNet  MATH  Google Scholar 

  14. Dimitrov D, Dvořák T, Gregor P, Skrekovski R (2009) Gray codes avoiding matchings. Discret Math Theor Comput Sci 11:123–148

    MathSciNet  MATH  Google Scholar 

  15. Dvořák T (2005) Hamiltonian cycles with prescribed edges in hypercubes. SIAM J Discret Math 19:135–144

    Article  MathSciNet  MATH  Google Scholar 

  16. Dvořák T, Gregor P (2007) Hamiltonian fault-tolerance of hypercubes. Electron Notes Discret Math 29:471–477

    Article  MATH  Google Scholar 

  17. Dvořák T, Gregor P (2008) Partitions of faulty hypercubes into paths with prescribed endvertices. SIAM J Discret Math 22(4):1448–1461

    Article  MathSciNet  MATH  Google Scholar 

  18. Gregor P, Dvořák T (2008) Path partitions of hypercubes. Inf Process Lett 108:402–406

    Article  MathSciNet  MATH  Google Scholar 

  19. Gros L (1872) Theorie du Baguenodier. Aime Vingtrinier, Lyon

    Google Scholar 

  20. Havel I (1984) On Hamiltonian circuits and spanning trees of hypercube. Cas Pest Mat 109:135–152

    MathSciNet  MATH  Google Scholar 

  21. Jo S, Park JH, Chwa KY (2013) Paired many-to-many disjoint path covers in faulty hypercubes. Theoret Comput Sci 513:1–24

    Article  MathSciNet  MATH  Google Scholar 

  22. Latifi S, Zheng SQ, Bagherzadeh N (1992) Optimal ring embedding in hypercubes with faulty links. In: Proceedings of the IEEE Symposium on Fault-Tolerant Computing, pp 178–184

  23. Lewinter M, Widulski W (1997) Hyper-Hamilton laceable and caterpillar-spannable product graphs. Comput Math Appl 34:99–104

    Article  MathSciNet  MATH  Google Scholar 

  24. Li XJ, Liu B, Ma MJ, Xu JM (2017) Many-to-many disjoint paths in hypercubes with faulty vertices. Discret Appl Math 217:229–242

    Article  MathSciNet  MATH  Google Scholar 

  25. Liu D, Li J (2010) Many-to-many \(n\)-disjoint path covers in \(n\)-dimensional hypercubes. Inf Process Lett 110(14–15):580–584

    Article  MathSciNet  MATH  Google Scholar 

  26. Liu JJ, Wang YL (2014) Hamiltonian cycles in hypercubes with faulty edges. Inf Sci 256:225–233

    Article  MathSciNet  MATH  Google Scholar 

  27. Lü HZ, Zhang HP (2014) Hyper-Hamiltonian laceability of balanced hypercubes. J Supercomput 68:302–314

    Article  Google Scholar 

  28. Park JH (2016) Unpaired many-to-many disjoint path covers in restricted hypercube-like graphs. Theoret Comput Sci 617:45–64

    Article  MathSciNet  MATH  Google Scholar 

  29. Park JH, Kim HC, Lim HS (2006) Many-to-many disjoint path covers in hypercube-like interconnection networks with faulty elements. IEEE Trans Parallel Distrib Syst 17(3):227–240

    Article  Google Scholar 

  30. Shih YK, Kao SS (2011) One-to-one disjoint path covers on \(k\)-ary \(n\)-cubes. Theoret Comput Sci 412:4513–4530

    Article  MathSciNet  MATH  Google Scholar 

  31. Tsai CH (2004) Linear array and ring embeddings in conditional faulty hypercubes. Theoret Comput Sci 314:431–443

    Article  MathSciNet  MATH  Google Scholar 

  32. Tsai CH, Tan JM, Liang T, Hsu LH (2002) Fault tolerant Hamiltonian laceability of hypercubes. Inf Process Lett 83:301–306

    Article  MathSciNet  MATH  Google Scholar 

  33. Valafar H, Arabnia HR, Williams G (2004) Distributed global optimization and its development on the multiring network. Int J Neural Parallel Sci Comput 12(4):465–490

    MathSciNet  MATH  Google Scholar 

  34. Wani MA, Arabnia HR (2003) Parallel edge-region-based segmentation algorithm targeted at reconfigurable multi-ring network. J Supercomput 25(1):43–63

    Article  MATH  Google Scholar 

  35. Xiang Y, Stewart I, Madelaine F (2011) Node-to-node disjoint paths in \(k\)-ary \(n\)-cube with faulty edges. In: Proceedings of IEEE 17th International Conference on Parallel and Distributed Systems, pp 181–187

  36. Zhang S, Wang S (2013) Many-to-many disjoint path covers in \(k\)-ary \(n\)-cubes. Theoret Comput Sci 491:103–118

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to express their gratitude to the anonymous reviewers for their kind suggestions on the original manuscript.

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Authors and Affiliations

  1. School of Sciences, Nanchang University, Nanchang, 330000, Jiangxi, People’s Republic of China

    Fan Wang

  2. Institute for Interdisciplinary Research, Jianghan University, Wuhan, 430056, Hubei, People’s Republic of China

    Weisheng Zhao

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  1. Fan Wang
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  2. Weisheng Zhao
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Correspondence to Fan Wang.

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This work is supported by NSFC (Grant Nos. 11501282 and 11861032).

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Wang, F., Zhao, W. One-to-one disjoint path covers in hypercubes with faulty edges. J Supercomput 75, 5583–5595 (2019). https://doi.org/10.1007/s11227-019-02817-6

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