Regular Factors Containing a Given Hamiltonian Cycle

  • Haruhide Matsuda
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3330)


Let k ≥ 1 be an integer and let G be a graph having a sufficiently large order n. Suppose that kn is even, the minimum degree of G is at least k + 2, and the degree sum of each pair of nonadjacent vertices in G is at least n + α, where α = 3 for odd k and α = 4 for even k. Then G has a k – factor (i.e. a k – regular spanning subgraph) which is edge-disjoint from a given Hamiltonian cycle. The lower bound on the degree condition is sharp. As a consequence, we have an Ore-type condition for graphs to have a k – factor containing a given Hamiltonian cycle.


Minimum Degree Hamiltonian Cycle Disjoint Subset Large Order Span Subgraph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Cai, M., Li, Y., Kano, M.: A [k,k + 1]-factor containing a given Hamiltonian cycle. Science in China (ser. A) 41, 933–938 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Niessen, T.: A Fan-type result for regular factors. Ars Combinatoria 46, 277–285 (1997)zbMATHMathSciNetGoogle Scholar
  3. 3.
    Iida, T., Nishimura, T.: An Ore-type condition for the existence of k-factors in graphs. Graphs and Combinatorics 7, 353–361 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Ore, O.: Note on Hamilton circuits. Amer. Math. Monthly 67, 55 (1960)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Tutte, W.T.: The factorization of linear graphs. J. London Math. Soc. 22, 107–111 (1947)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Tutte, W.T.: The factors of graphs. Canad. J. Math. 4, 314–328 (1952)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Matsuda, H.: Degree conditions for the existence of [k,k + 1]-factors containing a given Hamiltonian cycle. Australasian journal of combinatorics 26, 273–281 (2002)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Matsuda, H.: Degree conditions for Hamiltonian graphs to have [a,b]-factors containing a given Hamiltonian cycle. Discrete Mathematics 280, 241–250 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Shi, M., Yuan, X., Cai, M., Favaron, O.: (3,k)-factor-critical graphs and toughness. Graphs and Combinatorics 15, 463–471 (1999)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Haruhide Matsuda
    • 1
  1. 1.Department of General EducationKyushu Tokai UniversityKumamotoJapan

Personalised recommendations