Skip to main content
Springer Nature Link
Log in

Cycle Spectra of Contraction-Critically 4-Connected Planar Graphs

  • Original Paper
  • Published:
Graphs and Combinatorics Aims and scope Submit manuscript

Abstract

Motivated by the long-standing and wide open pancyclicity conjectures of Bondy and Malkevitch, we study the cycle spectra of contraction-critically 4-connected planar graphs. We show that every contraction-critically 4-connected planar graph on n vertices contains a cycle of length k for every \(k \in \big \{\lfloor \frac{n}{2} \rfloor - \lceil \frac{n}{108} \rceil , \dots , \lfloor \frac{n}{2} \rfloor + \lfloor \frac{n}{36} \rfloor \big \} \cup \big \{\frac{2}{3} n, \dots , n\big \}\).

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Availability of Data and Material

Not applicable.

Code Availability

Not applicable.

Notes

  1. A square of a cycle is obtained from the cycle by joining each pair of vertices at distance two with an edge.

  2. A graph G is cyclically 4-edge-connected if, for every edge-cut S of G with less than 4 edges, \(G - S\) has a component that contains no cycle.

References

  1. Bondy, J.A.: Pancyclic graphs I. J. Combin. Theory Ser. B 11(1), 80–84 (1971)

    Article  MathSciNet  Google Scholar 

  2. Bondy, J.A.: Pancyclic graphs: recent results. In: Hajnal, A., Rado, R., Sós, V. (eds.) Infinite and Finite Sets (Colloq. Keszthely, 1973; Dedicated to P. Erdős on his 60th Birthday), Volume10 of Colloq. Math. Soc. János Bolyai, pp. 181–187. North-Holland, Amsterdam (1975)

    Google Scholar 

  3. Chen, G., Fan, G., Yu, X.: Cycles in 4-connected planar graphs. Eur. J. Combin. 25, 763–780 (2004)

    Article  MathSciNet  Google Scholar 

  4. Cui, Q., Hu, Y., Wang, J.: Long cycles in 4-connected planar graphs. Discrete Math. 309(5), 1051–1059 (2009)

    Article  MathSciNet  Google Scholar 

  5. Fijavž, G., Juvan, M., Mohar, B., Škrekovski, R.: Planar graphs without cycles of specific lengths. Eur. J. Combin. 23(4), 377–388 (2002)

    Article  MathSciNet  Google Scholar 

  6. Fontet, M.: Graphes 4-essentiels. C. R. Acad. Sci. Paris 287, 289–290 (1978)

    MathSciNet  MATH  Google Scholar 

  7. Kriesell, M.: A survey on contractible edges in graphs of a prescribed vertex connectivity. Graphs Combin. 18, 1–30 (2002)

    Article  MathSciNet  Google Scholar 

  8. Lo, O.-H.S.: Find subtrees of specified weight and cycles of specified length in linear time. J Graph Theory (To appear)

  9. Malkevitch, J.: On the lengths of cycles in planar graphs. In: Capobianco, M., Frechen, J., Krolik, M. (eds.) Recent Trends in Graph Theory (Proc. Conf., New York 1970) Volume 186 of Lecture Notes in Mathematics, pp. 191–195. Springer, Berlin (1971)

    Google Scholar 

  10. Malkevitch, J.: Cycle lengths in polytopal graphs. In: Alavi, Y., Lick, D. (eds.) Theory and Applications of Graphs (Proc. Conf., Michigan 1976) Volume 642 of Lecture Notes in Computer Science, pp. 364–370. Springer, Berlin (1978)

    Google Scholar 

  11. Malkevitch, J.: Polytopal graphs. In: Beineke, L., Wilson, R. (eds.) Selected Topics in Graph Theory, vol. 3, pp. 169–188. Academic Press, New York (1988)

    MATH  Google Scholar 

  12. Martinov, N.: Uncontractible 4-connected graphs. J. Graph Theory 6, 343–344 (1982)

    Article  MathSciNet  Google Scholar 

  13. Plummer, M.D.: Problems. In: Hajnal, A., Rado, R., Sós, V. (eds.) Infinite and Finite Sets (Colloq. Keszthely, 1973; Dedicated to P. Erdős on his 60th Birthday), Volume10 of Colloq. Math. Soc. János Bolyai, pp. 1549–1550. North-Holland, Amsterdam (1975)

    Google Scholar 

  14. Sanders, D.P.: On Hamilton cycles in certain planar graphs. J. Graph Theory 21(1), 43–50 (1996)

    Article  MathSciNet  Google Scholar 

  15. Thomas, R., Yu, X.: 4-connected projective-planar graphs are Hamiltonian. J. Combin. Theory Ser. B 62, 114–132 (1994)

    Article  MathSciNet  Google Scholar 

  16. Thomassen, C.: A theorem on paths in planar graphs. J. Graph Theory 7(2), 169–176 (1983)

    Article  MathSciNet  Google Scholar 

  17. Tutte, W.T.: A theorem on planar graphs. Trans. Am. Math. Soc. 82, 99–116 (1956)

    Article  MathSciNet  Google Scholar 

  18. Wang, W., Lih, K.-W.: Choosability and edge choosability of planar graphs without 5-cycles. Appl. Math. Lett. 15(5), 561–565 (2002)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors would like to thank three anonymous referees for their careful reading and helpful comments which greatly improved the presentation of this paper; and to thank Matthias Kriesell and Tomáš Madaras for bringing our attention to this research problem and for inspiring discussions.

Funding

On-Hei Solomon Lo’s research was partially supported by NSFC Grants 11971406 and 11622110. Jens M. Schmidt’s research was partially supported by the Grant SCHM 3186/2-1 (401348462) from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation).

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Xiamen University, Xiamen, 361005, People’s Republic of China

    On-Hei Solomon Lo

  2. School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, People’s Republic of China

    On-Hei Solomon Lo

  3. Hamburg University of Technology, Institute for Algorithms and Complexity, Hamburg, Germany

    Jens M. Schmidt

Authors
  1. On-Hei Solomon Lo
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Jens M. Schmidt
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to On-Hei Solomon Lo.

Ethics declarations

Conflict of interest

Not applicable.

Additional information

Publisher's Note

Springer Nature 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

Lo, OH.S., Schmidt, J.M. Cycle Spectra of Contraction-Critically 4-Connected Planar Graphs. Graphs and Combinatorics 37, 2129–2137 (2021). https://doi.org/10.1007/s00373-021-02358-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00373-021-02358-x

Keywords