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Leaf number and Hamiltonian \(C_4\)-free graphs

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Abstract

Let G be a simple 2-connected, \(C_4\)-free graph with minimum degree \(\delta (G)\ge 4\) and leaf number L(G) such that \(\delta (G)\ge \displaystyle \frac{1}{2}L(G)\). We show that G is Hamiltonian. In addition, we provide family of graphs to show that the results are best possible for aforementioned class of graphs. The results, apart from supporting the conjecture (Graffiti.pc 190) of the computer program Graffiti.pc, instructed by DeLaVi\({\tilde{n}}\)a, provide a new sufficient condition for Hamiltonicity in \(C_4\)-free graphs.

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References

  1. Fernandes, L.M., Gouvea, L.: Spanning trees with a constraint on the number of leaves. Eur. J. Oper. Res. 1104, 250–261 (1998)

    Article  Google Scholar 

  2. Mukwembi, S.: Minimum degree, leaf number and traceability. CMJ 63(138), 539–545 (2013)

    MATH  MathSciNet  Google Scholar 

  3. Goodman, S., Hedetniemi, S.: Sufficient conditions for a graph to be Hamiltonian. J. Comb. Theory Ser. B 16, 175–180 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  4. Storer, J.A.: Constructing full spanning trees for cubic graphs. Inf. Process Lett 13, 8–11 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  5. Mukwembi, S.: Minimum degree, leaf number, and Hamiltonicity. Am. Math. Mon. 120(02), 115 (2013)

    Article  MATH  Google Scholar 

  6. Mukwembi, S.: On spanning cycles, paths and trees. Discret. Appl. Math. 161, 2217–2222 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  7. DeLavina, E., Waller, B.: Spanning trees with many leaves and average distance. Electron. J. Comb. 15, 1–16 (2008)

    MATH  MathSciNet  Google Scholar 

  8. Kleitman, D.J., West, D.B.: Spanning trees with many leaves. SIAM J. Discret. Math. 4, 99–106 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  9. Griggs, J.R., Wu, M.: Spanning trees in graphs of minimum degree 4 or 5. Discret. Math. 104, 167–183 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  10. DeLavina, E.: Written on the wall II (conjectures of Graffiti.pc), vol. 104, pp. 167–183. http://cms.dt.uh.edu/faculty/delavinae/research/wowII/

  11. Dirac, G.A.: Some theorems on abstract graphs. Proc. Lond. Math. Soc. 2, 69–81 (1952)

    Article  MATH  MathSciNet  Google Scholar 

  12. Broersma, H.J., van den Heuvel, J., Veldman, H.J.: A generalisation of Ore’s theorem involving neighbourhood unions. Discret. Math. 122, 37–49 (1993)

    Article  MATH  Google Scholar 

  13. Ren, S.: A sufficient condition for graphs with large neighbourhoods unions to be traceable. Discret. Math. 161, 229–234 (1996)

    Article  MATH  Google Scholar 

  14. Li, H.: Hamiltonian cycles in \(2\)-connected claw free graphs. J. Graph Theory. 20, 447–457 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  15. Hilbig, F.: Kantenstruckturen in Nichthamiltonschen Graphen (PHD Thesis). Technische Universitat (1986)

  16. Jackson, B.: Neighbourhood unions and Hamilton cycles. J. Gr. Theory. 15, 443–451 (1991)

    Article  MATH  Google Scholar 

  17. Fraudree, R., Gould, R.J., Jacobson, M.S., Lesniak, L.M.: Neigbhourhood unions and Hamilton properties in graphs. J. Comb. Theory B 46, 1–20 (1989)

    Article  Google Scholar 

  18. Ore, O.: Note on Hamiltonian circuits. Am. Math. Mon. 67, 55 (1960)

    Article  MATH  Google Scholar 

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Acknowledgements

We gratefully acknowledge financial support by the DAAD.

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  1. Department of Mathematics, University of Zimbabwe, Box MP 167, Mount Pleasant, Harare, Zimbabwe

    P. Mafuta

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  1. P. Mafuta
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Mafuta, P. Leaf number and Hamiltonian \(C_4\)-free graphs. Afr. Mat. 28, 1067–1074 (2017). https://doi.org/10.1007/s13370-017-0503-y

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