Skip to main content
SpringerLink
Log in

Matchings extend to Hamiltonian cycles in hypercubes with faulty edges

  • Research Article
  • Published:
Frontiers of Mathematics in China Aims and scope Submit manuscript

Abstract

We consider the problem of existence of a Hamiltonian cycle containing a matching and avoiding some edges in an n-cube Qn; and obtain the following results. Let n ⩾ 3; M ⊂ E(Qn); and F ⊂ E(Qn)M with 1 ⩽ |F| ⩽ 2n − 4 − |M|: If M is a matching and every vertex is incident with at least two edges in the graph Qn − F; then all edges of M lie on a Hamiltonian cycle in Qn − F: Moreover, if |M| = 1 or |M| = 2; then the upper bound of number of faulty edges tolerated is sharp. Our results generalize the well-known result for |M| = 1

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bondy J A, Murty U S R. Graph Theory. New York: Springer, 2007

    MATH  Google Scholar 

  2. Caha R, Koubek V. Hamiltonian cycles and paths with a prescribed set of edges in hypercubes and dense sets. J Graph Theory, 2006, 51: 137–169

    Article  MathSciNet  Google Scholar 

  3. Chen X -B. Cycles passing through prescribed edges in a hypercube with some faulty edges. Inform Process Lett, 2007, 104: 211–215

    Article  MathSciNet  Google Scholar 

  4. Chen X -B. Cycles passing through a prescribed path in a hypercube with faulty edges. Inform Process Lett, 2010, 110: 625–629

    Article  MathSciNet  Google Scholar 

  5. Chen X -B. The 2-path-bipanconnectivity of hypercubes. Inform Sci, 2013, 239: 283–293

    Article  MathSciNet  Google Scholar 

  6. Chen X -B. Fault-free Hamiltonian cycles passing through a prescribed linear forest in 3-ary n-cube with faulty edges. Front Math China, 2014, 9: 17–30

    Article  MathSciNet  Google Scholar 

  7. Dvořák T. Hamiltonian cycles with prescribed edges in hypercubes. SIAM J Discrete Math, 2005, 19: 135–144

    Article  MathSciNet  Google Scholar 

  8. Dvořák T, Greror P. Hamiltonian paths with prescribed edges in hypercubes. Discrete Math, 2007, 307: 1982–1998

    Article  MathSciNet  Google Scholar 

  9. Fink J. Perfect matchings extend to Hamiltonian cycles in hypercubes. J Combin Theory Ser B 2007, 97: 1074–1076

    Article  MathSciNet  Google Scholar 

  10. Li T K, Tsai C H, Tan J J M, Hsu L H. Bipanconnectivity and edge-fault-tolerant bipancyclicity of hypercubes. Inform Process Lett, 2003, 87: 107–110

    Article  MathSciNet  Google Scholar 

  11. Stewart T A. Hamiltonian cycles through prescribed edges in k-ary n-cubes. In: Combinatorial Optimization and Applications. Lecture Notes in Comput Sci, Vol 6831. 2011, 82–97

    Article  MathSciNet  Google Scholar 

  12. Tsai C H. Linear array and ring embeddings in conditional faulty hypercubes. Theoret Comput Sci, 2004, 314: 431–443

    Article  MathSciNet  Google Scholar 

  13. Tsai C H. Fault-free cycles passing through prescribed paths in hypercubes with faulty edges. Appl Math Lett, 2009, 22: 852–855

    Article  MathSciNet  Google Scholar 

  14. Wang F, Sun W. Perfect matchings extended to two or more Hamiltonian cycles in hypercubes. Discrete Math, 2019, 342: 505–510

    Article  MathSciNet  Google Scholar 

  15. Wang F, Zhang H. Prescribed matchings extended to Hamiltonian cycles in hypercubes with faulty edges. Discrete Math, 2014, 321: 35–44

    Article  MathSciNet  Google Scholar 

  16. Wang F, Zhang H. Matchings extend to Hamiltonian cycles in k-ary n-cubes. Inform Sci, 2015, 305: 1–13

    Article  MathSciNet  Google Scholar 

  17. Wang S, Li J, Wang R. Hamiltonian paths and cycles with prescribed edges in the 3-ary n-cubes. Inform Sci, 2011, 181: 3054–3065

    Article  MathSciNet  Google Scholar 

  18. Wang S, Li J, Wang R. Hamiltonian cycles passing through linear forests in k-ary n-cubes. Discrete Appl Math, 2011, 159: 1425–1435

    Article  MathSciNet  Google Scholar 

  19. Wang W Q, Chen X -B. A fault-free Hamiltonian cycle passing through prescribed edges in a hypercube with faulty edges. Inform Process Lett, 2008, 107: 205–210

    Article  MathSciNet  Google Scholar 

  20. Xu J -M, Ma M. A survey on path and cycle embedding in some networks. Front Math China, 2009, 4: 217–252

    Article  MathSciNet  Google Scholar 

  21. Yang Y, Li J, Wang S. Embedding fault-free Hamiltonian paths with prescribed linear forests into faulty ternary n-cubes. Theoret Comput Sci, 2019, 767: 1–15

    Article  MathSciNet  Google Scholar 

  22. Yang Y, Wang S. Fault-free Hamiltonian cycles passing through a linear forest in ternary n-cubes with faulty edges. Theoret Comput Sci, 2013, 491: 78–82

    Article  MathSciNet  Google Scholar 

  23. Zhang S, Zhang X. Fault-free Hamiltonian cycles passing through prescribed edges in k-ary n-cubes with faulty edges. IEEE Trans Parallel Distrib Syst, 2015, 26: 434–443

    Article  Google Scholar 

Download references

Acknowledgements

The work was supported by the National Natural Science Foundation of China (Grant No. 11401290).

Author information

Authors and Affiliations

  1. College of Mathematics and Statistics, Minnan Normal University, Zhangzhou, 363000, China

    Xie-Bin Chen

Authors
  1. Xie-Bin Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xie-Bin Chen.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chen, XB. Matchings extend to Hamiltonian cycles in hypercubes with faulty edges. Front. Math. China 14, 1117–1132 (2019). https://doi.org/10.1007/s11464-019-0810-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11464-019-0810-8

Keywords

MSC

Search

Navigation