Advertisement

Graphs and Combinatorics

, Volume 35, Issue 4, pp 827–836 | Cite as

Hamiltonian Spectra of Graphs

  • Li-Da Tong
  • Hao-Yu Yang
  • Xuding ZhuEmail author
Original Paper
  • 26 Downloads

Abstract

A hamiltonian walk in a digraph D is a closed spanning directed walk of D with minimum length. The length of a hamiltonian walk in D is called the hamiltonian number of D, and is denoted by h(D). The hamiltonian spectrum \(S_h(G)\) of a graph G is the set \(\{h(D): D\) is a strongly connected orientation of \(G\}\). In this paper, we present necessary and sufficient conditions for a graph G of order n to have \(S_h(G)=\{n\}\), \(\{n+1\}\), or \(\{n+2\}\). Then we construct some 2-connected graphs of order n with hamiltonian spectrum being a singleton \(n+k\) for some \(k\ge 3\), and graphs with their hamiltonian spectra being sets of consecutive integers.

Keywords

Hamiltonian spectrum Hamiltonian number Orientation 

Mathematics Subject Classification

05C45 05C38 05C69 

Notes

Acknowledgements

We thank the referee for a careful reading of the manuscript and valuable comments.

References

  1. 1.
    Asano, T., Nishizeki, T., Watanabe, T.: An upper bound on the length of a hamiltonian walk of a maximai planar graph. J. Graph Thoery 4, 315–336 (1980)CrossRefzbMATHGoogle Scholar
  2. 2.
    Asano, T., Nishizeki, T., Watanabe, T.: An approximation algorithm for the hamiltonian walk problems on maximal planar graphs. Discrete Appl. Math. 5, 211–222 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bermond, J.C.: On hamiltonian walks. Congr. Numer. 15, 41–51 (1976)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Bondy, J.A., Chvátal, V.: A method in graph theory. Discrete Math. 15, 111–135 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bondy, J.A., Murty, U.S.R.: Graph Theory with Applications. American Elsevier Publishing Co., Inc., New York (1976)CrossRefzbMATHGoogle Scholar
  6. 6.
    Chang, G.J., Chang, T.-P., Tong, L.-D.: The hamiltonian numbers of Möbius double loop networks. J. Comb. Optim. 23, 462–470 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chartrand, G., Saenpholphat, V., Thomas, T., Zhang, P.: On the hamiltonian number of a graph. Congr. Numer. 165, 56–64 (2003)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Chartrand, G., Saenpholphat, V., Thomas, T., Zhang, P.: A new look at hamiltonian walks. Bull. ICA 42, 37–52 (2004)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Chang, T.P., Tong, L.-D.: The hamiltonian numbers in digraphs. J. Comb. Optim. 25, 694–701 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Goodman, S.E., Hedetniemi, S.T.: On hamiltonian walks in graphs. In: Proc. Fourth Southestern Conf. Combin., Graph Theory and Computing, Utilitas Math., vol. 1973, pp. 335–342 (1973)Google Scholar
  11. 11.
    Goodman, S.E., Hedetniemi, S.T.: On hamiltonian walks in graphs. SIAM J. Comput. 3, 214–221 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Kral, D., Tong, L.-D., Zhu, X.: Upper hamiltonian number and the hamiltonian spectra of graphs. Aust. J. Combin. 35, 329–340 (2006)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Thaithae, S., Punnim, N.: The Hamiltonian number of cubic graphs. Lect. Notes Comput. Sci. 4535, 213–223 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Thaithae, S., Punnim, N.: The Hamiltonian number of graphs with prescribed connectivity. ARS Combin. 90, 237–244 (2009)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Tong, L.-D., Yang, H.-Y.: Hamiltonian numbers in oriented graphs. J. Comb. Optim. 34, 1210–1217 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Vacek, P.: On open hamiltonian walks in graphs. Arch. Math. 27A, 105–111 (1991)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Vacek, P.: Bounds of lengths of open hamiltonian walks. Arch. Math. 28, 11–16 (1992)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Japan KK, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Applied MathematicsNational Sun Yat-sen UniversityKaohsiungTaiwan
  2. 2.Department of MathematicsZhejiang Normal UniversityJinhuaChina

Personalised recommendations