Skip to main content

The Completion of a Classification for Maximal Nonhamiltonian Burkard–Hammer Graphs

Abstract

A graph G=(V,E) is called a split graph if there exists a partition V=IK such that the subgraphs G[I] and G[K] of G induced by I and K are empty and complete graphs, respectively. Burkard and Hammer gave a necessary condition for a split graph G with |I|<|K| to be Hamiltonian (J. Comb. Theory, Ser. B 28:245–248, 1980). We will call a split graph G with |I|<|K| satisfying this condition a Burkard–Hammer graph. Further, a split graph G is called a maximal nonhamiltonian split graph if G is nonhamiltonian but G+uv is Hamiltonian for every \(uv\not\in E\), where uI and vK. N.D. Tan and L.X. Hung have classified maximal nonhamiltonian Burkard–Hammer graphs G with minimum degree δ(G)≥|I|−3. Recently, N.D. Tan and Iamjaroen have classified maximal nonhamiltonian Burkard–Hammer graphs with |I|≠6,7 and δ(G)=|I|−4. In this paper, we complete the classification of maximal nonhamiltonian Burkard–Hammer graphs with δ(G)=|I|−4 by finding all such graphs for the case |I|=6,7.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3

References

  1. Behzad, M., Chartrand, G.: Introduction to the Theory of Graphs. Allyn & Bacon, Boston (1971)

    MATH  Google Scholar 

  2. Burkard, R.E., Hammer, P.L.: A note on Hamiltonian split graphs. J. Comb. Theory, Ser. B 28, 245–248 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  3. Chvátal, V., Hammer, P.L.: Aggregation of inequalities in integer programming. Ann. Discrete Math. 1, 145–162 (1977)

    Article  Google Scholar 

  4. Földes, S., Hammer, P.L.: Split graphs. In: Proceedings of the Eighth Southeastern Conference on Combinatorics, Graph Theory and Computing, Louisiana State Univ., Baton Rouge, LA, 1977. Congressus Numerantium, vol. XIX, pp. 311–315. Utilitas Math., Winnipeg (1977)

    Google Scholar 

  5. Földes, S., Hammer, P.L.: On a class of matroid-producing graphs. In: Combinatorics, Proc. Fifth Hungarian Colloq., Keszthely, 1976. Colloq. Math. Soc. Janós Bolyai 18, vol. 1, pp. 331–352. North-Holland, Amsterdam (1978)

    Google Scholar 

  6. Henderson, P.B., Zalcstein, Y.: A graph-theoretic characterization of the PV chunk class of synchronizing primitive. SIAM J. Comput. 6, 88–108 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  7. Hesham, A.H., Hesham, El.R.: Task allocation in distributed systems: a split graph model. J. Comb. Math. Comb. Comput. 14, 15–32 (1993)

    MATH  Google Scholar 

  8. Kratsch, D., Lehel, J., Müller, H.: Toughness, Hamiltonicity and split graphs. Discrete Math. 150, 231–245 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  9. Peemöller, J.: Necessary conditions for Hamiltonian split graphs. Discrete Math. 54, 39–47 (1985)

    MATH  MathSciNet  Google Scholar 

  10. Peled, U.N.: Regular Boolean functions and their polytope. Chap. VI, Ph.D. Thesis, Univ. Waterloo, Dept. Comb. Optim. (1975)

  11. Tan, N.D.: A note on maximal nonhamiltonian Burkard–Hammer graphs. Vietnam J. Math. 34, 397–409 (2006)

    MATH  MathSciNet  Google Scholar 

  12. Tan, N.D., Hung, L.X.: Hamilton cycles in split graphs with large minimum degree. Discuss. Math. Graph Theory 24, 23–40 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  13. Tan, N.D., Hung, L.X.: On the Burkard–Hammer condition for Hamiltonian split graphs. Discrete Math. 296, 59–72 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  14. Tan, N.D., Iamjaroen, C.: Constructions for nonhamiltonian Burkard–Hammer graphs. In: Combinatorial Geometry and Graph Theory (Proc. of Indonesia–Japan Joint Conf.), Bandung, Indonesia, September 13–16, 2003. Lect. Notes Comput. Sci., vol. 3330, pp. 185–199. Springer, Berlin (2005)

    Chapter  Google Scholar 

  15. Tan, N.D., Iamjaroen, C.: A necessary condition for maximal nonhamiltonian Burkard–Hammer graphs. J. Discrete Math. Sci. Cryptogr. 9, 235–252 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  16. Tan, N.D., Iamjaroen, C.: A classification for maximal nonhamiltonian Burkard–Hammer graphs. Discuss. Math. Graph Theory 28, 67–89 (2008)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgements

I would like to express my sincere thanks to the referee for his valuable remarks which helped me to improve the paper.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Ngo Dac Tan.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Tan, N.D. The Completion of a Classification for Maximal Nonhamiltonian Burkard–Hammer Graphs. Viet J Math 41, 465–505 (2013). https://doi.org/10.1007/s10013-013-0046-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10013-013-0046-y

Keywords

  • Split graph
  • Burkard–Hammer condition
  • Burkard–Hammer graph
  • Hamiltonian graph
  • Maximal nonhamiltonian split graph

Mathematics Subject Classification (2000)

  • 05C45
  • 05C75