Induced Nets and Hamiltonicity of Claw-Free Graphs

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, access via your institution.

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

References

  1. 1.

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

  2. 2.

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

    MathSciNet  Article  Google Scholar 

  3. 3.

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

    MathSciNet  Google Scholar 

  4. 4.

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

    MathSciNet  Article  Google Scholar 

  5. 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)

    MathSciNet  Article  Google Scholar 

  6. 6.

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

    Google Scholar 

  7. 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. 8.

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

    MathSciNet  Article  Google Scholar 

  9. 9.

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

    MathSciNet  Article  Google Scholar 

  10. 10.

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

    MathSciNet  Article  Google Scholar 

  11. 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. 12.

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

    MathSciNet  Article  Google Scholar 

  13. 13.

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

    MathSciNet  Article  Google Scholar 

  14. 14.

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

    MathSciNet  Article  Google Scholar 

  15. 15.

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

    MathSciNet  Article  Google Scholar 

Download references

Author information

Affiliations

Authors

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

Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

Keywords

  • Hamiltonian cycles
  • Claw-free graphs
  • Induced nets

Mathematics Subject Classification

  • 05C45
  • 05C38
  • 05C75