Skip to main content
SpringerLink
Log in

Chvátal–Erdős Theorem: Old Theorem with New Aspects

  • Conference paper
Computational Geometry and Graph Theory (KyotoCGGT 2007)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 4535))

Abstract

The Chvátal–Erdős Theorem states that a 2-connected graph is hamiltonian if its independence number is bounded from above by its connectivity. In this short survey, we explore the recent development on the extensions and the variants of this theorem.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Amar, D., Fournier, I., Germa, A.: Pancyclism in Chvátal–Erdős graphs. Graphs Combin. 7, 101–112 (1991)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  3. Bondy, J.A.: A remark on two sufficient conditions for Hamilton cycles. Discrete Math. 22, 191–194 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  4. Brandt, S., Chen, G., Faudree, R., Gould, R.J., Lesniak, L.: Degree conditions for 2-factors. J. Graph Theory 24, 165–173 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  5. Broersma, H.J.: Existence of Δ λ -cycles and Δ λ -paths. J. Graph Theory 12, 499–507 (1988)

    Article  MathSciNet  Google Scholar 

  6. Chartrand, G., Lesniak, L.: Graphs & Digraphs, 3rd edn., pp. 317–327. Wadsworth & Brooks/Cole, Monterey (1996)

    MATH  Google Scholar 

  7. Chen, G., Gould, R.J., Kawarabayashi, K., Ota, K., Saito, A., Schiermeyer, I.: The Chvátal–Erdős condition and 2-factor with a specified number of components. Discuss. Math. Graph Theory 27, 401–407 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chvátal, V., Erdős, P.: A note on Hamilton circuits. Discrete Math. 2, 111–113 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  9. Egawa, Y.: Personal communication (2007)

    Google Scholar 

  10. Enomoto, H.: On the existence of disjoint cycles in a graph. Combinatorica. 18, 487–492 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  11. Enomoto, H., Kaneko, A., Saito, A., Wei, B.: Long cycles in triangle-free graphs with prescribed independence number and connectivity. J. Combin. Theory Ser. B 91, 43–55 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  12. Enomoto, H., Ozeki, K.: The independence number condition for the existence of a spanning f-tree, (preprint)

    Google Scholar 

  13. Flandrin, E., Li, H., Marczyk, A., Schiermeyer, I., Woźniak, M.: Chvátal–Erdős condition and pancyclism. Discuss. Math. Graph Theory 26, 335–342 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  14. Fouquet, J.L., Jolivet, J.L.: Problèmes combinatoires et théorie des graphes. problèmes Orsay 248 (1976)

    Google Scholar 

  15. Fournier, I.: Thèse d’Etat L.R.I., Université de Paris-Sud (1985)

    Google Scholar 

  16. Fournier, I.: Longest cycles in 2-connected graphs. Ann. Discrete Math. 27, 201–204 (1985)

    MATH  Google Scholar 

  17. Gould, R.J.: Updating the hamiltonian problem — a survey —. J. Graph Theory 15, 121–157 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  18. Fraisse, P.: A note on distance-dominating cycles. Discrete Math. 71, 89–92 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  19. Gould, R.J.: Advances on the hamiltonian problem — a survey —. Graphs Combin. 19, 7–52 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  20. Jackson, B., Ordaz, O.: Chvátal–Erdős conditions for paths and cycles in graphs and digraphs, a survey. Discrete Math. 84, 241–254 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kaneko, A., Yoshimoto, K.: A 2-factor with two components of a graph satisfying the Chvátal–Erdős condition. J. Graph Theory 43, 269–279 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kouider, M.: Cycles in graphs with prescribed stability number and connectivity. J. Combin. Theory Ser.B 60, 315–318 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  23. Lou, D.: The Chvátal–Erdős condition for cycles in triangle-free graphs. Discrete Math. 152, 253–257 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  24. Mannoussakis, Y.: Thèse de \(3^{\text{ieme}}\) cycle, L.R.I., Université de Paris-Sud (1985)

    Google Scholar 

  25. Matsuda, H., Matsumura, H.: On a k-tree containing specified leaves in a graph. Graphs Combin. 22, 371–381 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  26. Neumann-Lara, V., Rivera-Campo, E.: Spanning trees with bounded degrees. Combinatorica. 11, 55–61 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  27. Oberly, D.J., Sumner, D.P.: Every connected, locally connected nontrivial graph with induced claw is Hamiltonian. J. Graph Theory 3, 351–356 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  28. Ore, O.: Note on Hamilton circuits. Amer. Math. Monthly 67, 55 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  29. Saito, A., Yamashita, T.: Cycles within specified distance from each vertex. Discrete Math. 278, 219–226 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  30. Win, S.: On a conjecture of Las Vergnas concerning certain spanning trees in graphs. Resultate Math. 2, 215–224 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  31. Zhang, C.-Q.: Hamiltonian cycles in claw-free graphs. J. Graph Theory 12, 209–216 (1988)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, Nihon University, Sakurajosui 3–25–40, Setagaya-Ku, Tokyo, 156-8550, Japan

    Akira Saito

Authors
  1. Akira Saito
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. School of Informatics, Kyoto University, Yosida-Honmati, 606-8501, Kyoto, Japan

    Hiro Ito

  2. Ibaraki University, Nakanarusawa, Hitachi, 316-8511, Ibaraki, Japan

    Mikio Kano

  3. Department of Architecture and Architectural Engineering, Kyoto University, Nishikyo-ku, 615-8540, Kyoto, Japan

    Naoki Katoh

  4. Graduate School of Science, Department of Mathematics and Information Sciences, Osaka Prefecture University, 599-8531, Sakai, Japan

    Yushi Uno

Rights and permissions

Reprints and permissions

Copyright information

© 2008 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Saito, A. (2008). Chvátal–Erdős Theorem: Old Theorem with New Aspects. In: Ito, H., Kano, M., Katoh, N., Uno, Y. (eds) Computational Geometry and Graph Theory. KyotoCGGT 2007. Lecture Notes in Computer Science, vol 4535. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-89550-3_21

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-89550-3_21

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-89549-7

  • Online ISBN: 978-3-540-89550-3

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation