Hamiltonian Paths and Cycles in Planar Graphs

  • Sudip Biswas
  • Stephane Durocher
  • Debajyoti Mondal
  • Rahnuma Islam Nishat
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7402)


We examine the problem of counting the number of Hamiltonian paths and Hamiltonian cycles in outerplanar graphs and planar graphs, respectively. We give an O( n ) upper bound and an Ω(α n ) lower bound on the maximum number of Hamiltonian paths in an outerplanar graph with n vertices, where α ≈ 1.46557 is the unique real root of α 3 = α 2 + 1. For any positive integer n ≥ 6, we define an outerplanar graph G, called a ZigZag outerplanar graph, such that the number of Hamiltonian paths starting at a single vertex in G is the maximum over all possible outerplanar graphs with n vertices. Finally, we prove a 2.2134 n upper bound on the number of Hamiltonian cycles in planar graphs, which improves the previously best known upper bound 2.3404 n .


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bax, E.T.: Inclusion and exclusion algorithm for the Hamiltonian path problem. Information Processing Letters 47(4), 203–207 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Buchin, K., Knauer, C., Kriegel, K., Schulz, A., Seidel, R.: On the Number of Cycles in Planar Graphs. In: Lin, G. (ed.) COCOON 2007. LNCS, vol. 4598, pp. 97–107. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  3. 3.
    Collins, K.L., Krompart, L.B.: The number of Hamiltonian paths in a rectangular grid. Discrete Mathematics 169(1-3), 29–38 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Curran, S.J., Gallian, J.A.: Hamiltonian cycles and paths in Cayley graphs and digraphs - A survey. Discrete Mathematics 156(1-3), 1–18 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    de Mier, A., Noy, M.: On the maximum number of cycles in outerplanar and series-parallel graphs. Electronic Notes in Discrete Mathematics 34, 489–493 (2009)CrossRefGoogle Scholar
  6. 6.
    Eppstein, D.: The traveling salesman problem for cubic graphs. Journal of Graph Algorithms and Applications 11(1), 61–81 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Garey, M., Johnson, D.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman and Company (1979)Google Scholar
  8. 8.
    Gebauer, H.: Finding and enumerating Hamilton cycles in 4-regular graphs. Theoretical Computer Science 412(35), 4579–4591 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Pichler, R., Rümmele, S., Woltran, S.: Counting and Enumeration Problems with Bounded Treewidth. In: Clarke, E.M., Voronkov, A. (eds.) LPAR-16 2010. LNCS (LNAI), vol. 6355, pp. 387–404. Springer, Heidelberg (2010)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Sudip Biswas
    • 1
  • Stephane Durocher
    • 2
  • Debajyoti Mondal
    • 2
  • Rahnuma Islam Nishat
    • 3
  1. 1.Department of Computer ScienceLouisiana State UniversityUSA
  2. 2.Department of Computer ScienceUniversity of ManitobaCanada
  3. 3.Department of Computer ScienceUniversity of VictoriaCanada

Personalised recommendations