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Induced Nets and Hamiltonicity of Claw-Free Graphs

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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.

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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 

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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

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  1. Shuya Chiba
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  2. Jun Fujisawa
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Correspondence to Jun Fujisawa.

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

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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.

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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

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

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