Nothing Special   »   [go: up one dir, main page]
Skip to main content
Springer Nature Link
Log in

Induced Nets and Hamiltonicity of Claw-Free Graphs

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

Abstract

The connected graph of degree sequence 3, 3, 3, 1, 1, 1 is called a net, and the vertices of degree 1 in a net are called its endvertices. Broersma conjectured in 1993 that a 2-connected graph G with no induced \(K_{1,3}\) is hamiltonian if every endvertex of each induced net of G has degree at least \((|V(G)|-2)/3\). In this paper we prove this conjecture in the affirmative.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  1. Broersma, H.J.: Problem 2. Workshop Cycles and Colourings (Novy Smokovec, 1993). http://umv.science.upjs.sk/c&c/history/93problems.pdf

  2. Brousek, J.: Minimal \(2\)-connected non-Hamiltonian claw-free graphs. Discrete Math. 191, 57–64 (1998)

    Article  MathSciNet  Google Scholar 

  3. Catlin, P.A.: Supereulerian graphs, collapsible graphs, and four-cycles. Congr. Numer. 58, 233–246 (1987)

    MathSciNet  Google Scholar 

  4. Catlin, P.A.: A reduction method to find spanning Eulerian subgraphs. J. Graph Theory 12(1), 29–44 (1988)

    Article  MathSciNet  Google Scholar 

  5. Čada, R., Li, B., Ning, B., Zhang, S.: Induced subgraphs with large degrees at end-vertices for Hamiltonicity of claw-free graphs. Acta Math. Sin. 32(7), 845–855 (2016)

    Article  MathSciNet  Google Scholar 

  6. Diestel, R.: Graph Theory. Graduate Texts in Mathematics, vol. 173, 5th edn. Springer, Berlin (2017)

    Google Scholar 

  7. Duffus, D., Gould, R.J., Jacobson, R.J.: Forbidden Subgraphs and the Hamiltonian Theme. The Theory and Applications of Graphs (Kalamazoo, Mich., 1980). Wiley, New York, pp. 297–316 (1981)

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  10. Gould, R.J.: Recent advances on the Hamiltonian problem: survey III. Graphs Combin. 30, 1–46 (2014)

    Article  MathSciNet  Google Scholar 

  11. Harary, F., Nash-Williams, C.St.J.A.: On Eulerian and Hamiltonian graphs and line graphs. Can. Math. Bull. 8, 701–709 (1965)

  12. Lai, H.-J.: Graph whose edges are in small cycles. Discrete Math. 94, 11–22 (1991)

    Article  MathSciNet  Google Scholar 

  13. Matthews, M.M., Sumner, D.P.: Longest paths and cycles in \(K_{1,3}\)-free graphs. J. Graph Theory 9, 269–277 (1985)

    Article  MathSciNet  Google Scholar 

  14. Pfender, F.: Hamiltonicity and forbidden subgraphs in \(4\)-connected graphs. J. Graph Theory 49(4), 262–272 (2005)

    Article  MathSciNet  Google Scholar 

  15. Ryjáček, Z.: On a closure concept in claw-free graphs. J. Combin. Theory Ser. B 70(2), 217–224 (1997)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Applied Mathematics, Faculty of Advanced Science and Technology, Kumamoto University, 2-39-1, Kurokami, Kumamoto, 860-8555, Japan

    Shuya Chiba

  2. Faculty of Business and Commerce, Keio University, Hiyoshi 4-1-1, Kohoku-Ku, Yokohama, Kanagawa, 223-8521, Japan

    Jun Fujisawa

Authors
  1. Shuya Chiba
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jun Fujisawa
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jun Fujisawa.

Additional information

Dedicated to Professor Katsuhiro Ota on the occasion of his 60th birthday

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Shuya Chiba: work supported by Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (C) 17K05347 and 20K03720. Jun Fujisawa: work supported by Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (B) 16H03952, (C) 17K05349 and (C) 20K03723.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chiba, S., Fujisawa, J. Induced Nets and Hamiltonicity of Claw-Free Graphs. Graphs and Combinatorics 37, 663–690 (2021). https://doi.org/10.1007/s00373-020-02265-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00373-020-02265-7

Keywords

Mathematics Subject Classification

Search

Navigation