Acta Mathematica Hungarica

, Volume 152, Issue 1, pp 217–226 | Cite as

Hamiltonicity, minimum degree and leaf number

Article

Abstract

We prove a new sufficient condition for a connected graph to be Hamiltonian in terms of the leaf number and the minimum degree. Our results give solutions to conjectures on the Hamiltonicity and traceability of graphs. We considerably generalize known results in the area by showing that if G is a connected graph having minimum degree \({\delta}\) and leaf number L such that \({\delta \ge \frac{L}{2}+1}\), then G is Hamiltonian and thus traceable.

Key words and phrases

Hamiltonicity traceability minimum degree leaf number 

Mathematics Subject Classification

05C45 05C35 05C05 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    M. Ajtai, J. Komlíos and E. Szemeríedi, First occurrence of Hamilton cycles in random graphs, in: Cycles in Graphs (Burnaby, B.C., 1982), North-Holland Math. Stud., 115, North-Holland (Amsterdam, 1985), pp. 173–178.Google Scholar
  2. 2.
    Chen Y.-C., Füredi Z.: Hamiltonian Kneser graphs. Combinatorica, 22, 147–149 (2002)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    E. DeLaViña  Written on the Wall II (Conjectures of Graffiti.pc), http://cms.dt.uh.edu/faculty/delavinae/research/wowII.
  4. 4.
    Dirac G.A.: Some theorems on abstract graphs,. Proc. London Math. Soc. (3) 2, 69–81 (1952)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Erdős P., Hobbs A.M.: Hamiltonian cycles in regular graphs of moderate degree,. J. Combin. Theory Ser. B, 23, 139–142 (1977)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Fernandes L.M., Gouveia L.: Minimal spanning trees with a constraint on the number of leaves,. European J. Oper. Res., 104, 250–261 (1998)CrossRefMATHGoogle Scholar
  7. 7.
    Griggs J. R., Wu M.: Spanning trees in graphs of minimum degree 4 or 5,. Discrete Math., 104, 167–183 (1992)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Kleitman D.J., West D.B.: Spanning trees with many leaves,. SIAM J. Discrete Math., 4, 99–106 (1991)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    S. Mukwembi, Minimum degree, leaf number, and Hamiltonicity, Amer. Math. Monthly, 120 (2013), 115.Google Scholar
  10. 10.
    Mukwembi S.: Minimum degree, leaf number and traceability,. Czechoslovak Math. J., 63, 539–545 (2013)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    S. Mukwembi: On spanning cycles, paths and trees,. Discrete Appl. Math., 161, 2217–2222 (2013)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    C. St. J. A. Nash-Williams, Edge-disjoint hamiltonian circuits in graphs with vertices of large valency, in: Studies in Pure Mathematics (Presented to Richard Rado), Academic Press (London, 1971), pp. 157–183.Google Scholar
  13. 13.
    Síarközy G. N.: A fast parallel algorithm for finding Hamiltonian cycles in dense graphs,. Discrete Math. 309, 1611–1622 (2009)MathSciNetCrossRefGoogle Scholar
  14. 14.
    I. Stojmenoviíc, Topological properties of interconnection networks, in: Combinatorial Optimization in Communication Networks, M. X. Cheng, Y. Li, D. Du (Eds.), Kluwer Academic Publishers (Dordrecht, 2006), pp. 427–466.Google Scholar
  15. 15.
    G. Wiener, Leaf-critical and leaf-stable graphs, J. Graph Theory (in press).Google Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2017

Authors and Affiliations

  • P. Mafuta
    • 1
  • S. Mukwembi
    • 1
    • 2
  • S. Munyira
    • 1
  • T. Vetrík
    • 3
  1. 1.Department of MathematicsUniversity of ZimbabweHarareZimbabwe
  2. 2.School of Mathematics, Statistics and Computer ScienceUniversity of KwaZulu-NatalDurbanSouth Africa
  3. 3.Department of Mathematics and Applied MathematicsUniversity of the Free StateBloemfonteinSouth Africa

Personalised recommendations