Odd Graphs Are Prism-Hamiltonian and Have a Long Cycle

  • Felipe De Campos Mesquita
  • Letícia Rodrigues Bueno
  • Rodrigo De Alencar Hausen
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8392)


The odd graph O k is the graph whose vertices are all subsets with k elements of a set {1,…,2k + 1}, and two vertices are joined by an edge if the corresponding pair of k-subsets is disjoint. A conjecture due to Biggs claims that O k is hamiltonian for k ≥ 3 and a conjecture due to Lovász implies that O k has a hamiltonian path for k ≥ 1. In this paper, we show that the prism over O k is hamiltonian and that O k has a cycle with .625|V(O k )| vertices at least.


hamiltonian cycle prism over a graph odd graph 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Babai, L.: Long cycles in vertex-transitive graphs. Journal of Graph Theory 3(3), 301–304 (1979)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Biggs, N.: Some odd graph theory. Annals of the New York Academy of Sciences 319, 71–81 (1979)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Bueno, L.R., Faria, L., Figueiredo, C.M.H., Fonseca, G.D.: Hamiltonian paths in odd graphs. Applicable Analysis and Discrete Mathematics 3(2), 386–394 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Bueno, L.R., Horák, P.: On hamiltonian cycles in the prism over the odd graphs. Journal of Graph Theory 68(3), 177–188 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Chen, Y.C.: Triangle-free hamiltonian Kneser graphs. Journal of Combinatorial Theory Series B 89, 1–16 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Duffus, D.A., Kierstead, H.A., Snevily, H.S.: An explicit 1-factorization in the middle of the boolean lattice. Journal of Combinatorial Theory, Series A 65, 334–342 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Havel, I.: Semipaths in directed cubes. In: Fiedler, M. (ed.) Graphs and other Combinatorial Topics, pp. 101–108. Teubner-Texte Math., Teubner (1983)Google Scholar
  8. 8.
    Horák, P., Kaiser, T., Rosenfeld, M., Ryjaček, Z.: The prism over the middle-levels graph is hamiltonian. Order 22(1), 73–81 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Jackson, B., Wormald, N.C.: k-walks of graphs. Australasian Journal of Combinatorics 2, 135–146 (1990)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Johnson, J.R., Kierstead, H.A.: Explicit 2-factorisations of the odd graph. Order 21, 19–27 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Johnson, J.R.: Long cycles in the middle two layers of the discrete cube. Journal of Combinatorial Theory Series A 105(2), 255–271 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Kaiser, T., Ryjáček, Z., Král, D., Rosenfeld, M., Voss, H.-J.: Hamilton cycles in prisms. Journal of Graph Theory 56, 249–269 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W. (eds.) Complexity of Computer Computations, pp. 85–103. Plenum Press, New York (1972)CrossRefGoogle Scholar
  14. 14.
    Lovász, L.: Problem 11. In: Combinatorial Structures and their Applications. Gordon and Breach (1970)Google Scholar
  15. 15.
    Paulraja, P.: A characterization of hamiltonian prisms. Journal of Graph Theory 17, 161–171 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Savage, C.D., Winkler, P.: Monotone gray codes and the middle levels problem. J. Combin. Theory Ser. A 70(2), 230–248 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Shields, I., Savage, C.D.: A note on hamilton cycles in Kneser graphs. Bulletin of the Institute for Combinatorics and Its Applications 40, 13–22 (2004)zbMATHMathSciNetGoogle Scholar
  18. 18.
    Shields, I., Shields, B.J., Savage, C.D.: An update on the middle levels problem. Discrete Mathematics 309(17), 5271–5277 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Shimada, M., Amano, K.: A note on the middle levels conjecture. CoRR abs/0912.4564 (2011)Google Scholar
  20. 20.
    Čada, R., Kaiser, T., Rosenfeld, M., Ryjáček, Z.: Hamiltonian decompositions of prisms over cubic graphs. Discrete Mathematics 286, 45–56 (2004)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Felipe De Campos Mesquita
    • 1
  • Letícia Rodrigues Bueno
    • 1
  • Rodrigo De Alencar Hausen
    • 1
  1. 1.Universidade Federal do ABC (UFABC)Santo AndréBrazil

Personalised recommendations