Skip to main content
SpringerLink
Log in

On Hamiltonian Cycles in a 2-Strong Digraphs with Large Degrees and Cycles

  • PART II “DISCRETE MATHEMATICAL PROBLEMS OF PATTERN RECOGNITION”
  • Published:
Pattern Recognition and Image Analysis Aims and scope Submit manuscript

Abstract

In this note we prove: let D be a 2-strong digraph of order \(n\) such that its \(n - 1\) vertices have degrees at least \(n + k\) and the remaining vertex \(z\) has degree at least \(n - k - 4,\) where \(k\) is a nonnegative integer. If \(D\) contains a cycle of length at least \(n - k - 2\) passing through \(z,\) then \(D\) is Hamiltonian. This result is best possible in some sense.

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

REFERENCES

  1. J. Adamus, “A degree sum condition for hamiltonicity in balanced bipartite digraphs,” Graphs Combinatorics 33, 43–51 (2017). https://doi.org/10.1007/s00373-016-1751-6

    Article  MathSciNet  Google Scholar 

  2. J. Adamus, “On dominating pair for Hamiltonicity in balanced bipartite digraphs,” Discrete Math. 344, 112240 (2021).

  3. J. Adamus and L. Adamus, “A degree condition for cycles of maximum length in bipartite digraphs,” Discrete Math. 312, 1117–1122 (2012). https://doi.org/10.1016/j.disc.2011.11.032

    Article  MathSciNet  Google Scholar 

  4. J. Adamus, L. Adamus, and A. Yeo, “On the Meyniel condition for hamiltonicity in bipartite digraphs,” Discrete Math. Theor. Comput. Sci. 16, 293–302 (2014). https://doi.org/10.46298/dmtcs.1264

    Article  MathSciNet  Google Scholar 

  5. J. Bang-Jensen and G. Z. Gutin, Digraphs: Theory, Algorithms and Applications, Springer Monographs in Mathematics (Springer, 2000). https://doi.org/10.1007/978-1-84800-998-1

  6. J. Bang-Jensen, G. Gutin, and H. Li, “Sufficient conditions for a digraph to be Hamiltonian,” J. Graph Theory 22, 181–187 (1996). https://doi.org/10.1002/(SICI)1097-0118(199606)22:2%3C181::AID-JGT9%3E3.0.CO;2-J

    Article  MathSciNet  Google Scholar 

  7. J.-C. Bermond and C. Thomassen, “Cycles in digraphs—A survey,” J. Graph Theory 5, 1–43 (1981). https://doi.org/10.1002/jgt.3190050102

    Article  MathSciNet  Google Scholar 

  8. J. A. Bondy and C. Thomassen, “A short proof of Meyniel’s theorem,” Discrete Math. 19, 195–197 (1977). https://doi.org/10.1016/0012-365X(77)90034-6

    Article  MathSciNet  Google Scholar 

  9. S. Kh. Darbinyan, “Cycles of any length in digraph with large semi-degrees,” Dokl. Akad. Nauk Arm. SSR 75 (4), 147–152 (1982).

    Google Scholar 

  10. S. Kh. Darbinyan, “Disproof of a conjecture of Thomassen,” Dokl. Akad. Nauk Arm. SSR 76 (2), 51–54 (1983).

    MathSciNet  Google Scholar 

  11. S. Kh. Darbinyan, “Hamiltonian and strongly Hamilton-connected digraphs,” Dokl. Akad. Nauk Arm. SSR 91 (1), 3–6 (1990).

    MathSciNet  Google Scholar 

  12. S. Kh. Darbinyan, “A sufficient condition for a digraph to be Hamiltonian,” Dokl. Akad. Nauk Arm. SSR 91 (2), 57–59 (1990).

    MathSciNet  Google Scholar 

  13. S. Kh. Darbinyan, “A sufficient conditions for Hamiltonian cycles in bipartite digraphs,” Discrete Appl. Math. 258, 87–96 (2019). https://doi.org/10.1016/j.dam.2018.11.024

    Article  MathSciNet  Google Scholar 

  14. S. Kh. Darbinyan, “A new sufficient condition for a digraph to be Hamiltonian—A proof of Manoussakis Conjecture,” Discrete Math. Theor. Comput. Sci. 22, 6086 (2021). https://doi.org/10.23638/DMTCS-22-4-12

    Article  MathSciNet  Google Scholar 

  15. S. Kh. Darbinyan, “On an extension of the Ghouila–Houri theorem,” Math. Probl. Comput. Sci. 58, 20–31 (2022). https://doi.org/10.51408/1963-0089

    Article  MathSciNet  Google Scholar 

  16. H. Ge, C. Ye, and Sh. Zhang, “A survey on sufficient conditions for Hamiltonian cycles in bipartite digraphs,” Discontinuity, Nonlinearity, Complexity 1 (2), 55–62 (2020). https://doi.org/10.5890/JVTSD.2020.03.003

    Article  Google Scholar 

  17. A. Ghouila-Houri, “Une condition suffisante d’existence d’un circuit hamiltonien,” C. R. Acad. Sci. Paris, Ser. A-B 251, 495–497 (1960).

  18. M. K. Goldberg, L. P. Levitskaya, and L. M. Satanovskyi, “On one strengthening of the Ghouila–Houri theorem,” in Vichislitelnaya Matematika i Vichislitelnaya Teknika (1971), pp. 56–61.

  19. R. J. Gould, “Resent advances on the Hamiltonian problem: Survey III,” Graphs Combinatorics 30, 1–46 (2014). https://doi.org/10.1007/s00373-013-1377-x

    Article  MathSciNet  Google Scholar 

  20. R. Häggkvist and C. Thomassen, “On pancyclic digraphs,” J. Comb. Theory, Ser. B 20, 20–40 (1976). https://doi.org/10.1016/0095-8956(76)90063-0

    Article  MathSciNet  Google Scholar 

  21. D. Kühn and D. Osthus, “A survey on Hamilton cycles in directed graphs,” Eur. J. Combinatorics 33, 750–766 (2012). https://doi.org/10.1016/j.ejc.2011.09.030

    Article  MathSciNet  Google Scholar 

  22. Y. Manoussakis, “Directed Hamiltonian graphs,” J. Graph Theory 16, 51–59 (1992). https://doi.org/10.1002/jgt.3190160106

    Article  MathSciNet  Google Scholar 

  23. M. Meyniel, “Une condition suffisante d’existence d’un circuit hamiltonien dans un graphe oriente,” J. Comb. Theory, Ser. B 14, 137–147 (1973). https://doi.org/10.1016/0012-365X(77)90034-6

    Article  MathSciNet  Google Scholar 

  24. C. J. A. Nash-Williams, “Hamilton circuits in graphs and digraphs,” in The Many Facets of Graph Theory, Ed. by G. Chartrand and S. F. Kapoor, Lecture Notes in Mathematics, Vol. 110 (Springer, Berlin, 1969), pp. 237–243. https://doi.org/10.1007/BFb0060123

  25. C. Thomassen, “Long cycles in digraphs,” Proc. London Math. Soc. s3-42, 231–251 (1981).https://doi.org/10.1112/PLMS/S3-42.2.231

  26. R. Wang, “Extremal digraphs on Woodal-type condition for Hamiltonian cycles in balanced bipartite digraphs,” J. Graph Theory 97, 194–207 (2021).

    Article  MathSciNet  Google Scholar 

  27. R. Wang, “A note on dominating pair degree condition for Hamiltonian cycles in balanced bipartite digraphs,” Graphs Combinatorics 38, 13 (2022).https://doi.org/10.1007/s00373-021-02404-8

  28. R. Wang and L. Wu, “Cycles of many lengths in balanced bipartite digraphs on dominating and dominated degree condition,” Discussiones Math. Graph Theory (2021). https://doi.org/10.7151/dmgt.2442

  29. R. Wang, L. Wu, and W. Meng, “Extremal digraphs on Meyniel-type conditions for Hamiltonian cycles in balanced bipartite digraphs,” Discrete Math. Theor. Comput. Sci. 23, 5851 (2022). https://doi.org/10.46298/dmtcs.5851

    Article  MathSciNet  Google Scholar 

  30. D. R. Woodall, “Sufficient conditions for circuits in graphs,” Proceeding London Math. Soc. 24, 739–755 (1972). https://doi.org/10.1112/plms/53-24.4.739

    Article  MathSciNet  Google Scholar 

  31. Z.-B. Zhang, X. Zhang, and X. Wen, “Directed Hamilton cycles in digraphs and matching alternating Hamilton cycles in bipartite graphs,” SIAM J. Discrete Math.” 27, 274–289 (2013). https://doi.org/10.1137/110837188

    Article  MathSciNet  Google Scholar 

Download references

ACKNOWLEDGMENTS

I am grateful to Professor Gregory Gutin for motivating me to present the complete proof of Theorem 8. Also thanks to Dr. Parandzem Hakobyan for formatting the manuscript of this paper.

Funding

This work was supported by ongoing institutional funding. No additional grants to carry out or direct this particular research were obtained.

Author information

Authors and Affiliations

  1. Institute for Informatics and Automation Problems, National Academy of Sciences of Republic of Armenia, 0014, Yerevan, Armenia

    S. Kh. Darbinyan

Authors
  1. S. Kh. Darbinyan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. Kh. Darbinyan.

Ethics declarations

The author of this work declares that he has no conflicts of interest.

Additional information

Samvel Kh. Darbinyan (born on June 20, 1950) Candidate of Mathematics (Minsk, 1981), Associate Professor, Leading Researcher of “Institute for Informatics and Automation Problems of the National Academy of Sciences of the Republic of Armenia.” Areas of scientific interest: Graph theory (cycles and paths in digraphs), VLSI design. Author (coauthor) of more than 50 scientific works.

Publisher’s Note.

Pleiades Publishing remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Darbinyan, S.K. On Hamiltonian Cycles in a 2-Strong Digraphs with Large Degrees and Cycles. Pattern Recognit. Image Anal. 34, 62–73 (2024). https://doi.org/10.1134/S105466182401005X

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S105466182401005X

Keywords:

Search

Navigation