An Implicit Degree Condition for Cyclability in Graphs

  • Hao Li
  • Wantao Ning
  • Junqing Cai
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6681)


A vertex subset X of a graph G is said to be cyclable in G if there is a cycle in G containing all vertices of X. Ore [6] showed that the vertex set of G with cardinality n ≥ 3 is cyclable (i.e. G is hamiltonian) if the degree sum of any pair of nonadjacent vertices in G is at least n. Shi [8] and Ota [7] respectively generalized Ore’s result by considering the cyclability of any vertex subset X of G under Ore type condition. Flandrin et al. [4] in 2005 extended Shi’s conclusion under the condition called regional Ores condition. Zhu, Li and Deng [10] introduced the definition of implicit degrees of vertices. In this work, we generalize the result of Flandrin et al. under their type condition with implicit degree sums. More precisely, we obtain that X is cyclable in a k-connected graph G if the implicit degree sum of any pair of nonadjacent vertices u,v ∈ X i is at least the order of G, where each X i , i = 1,2, ⋯ ,k is a vertex subset of G and X = ∪  k i = 1 X i . In [10], the authors demonstrated that the implicit degree of a vertex is at least the degree of the vertex. Hence our result is better than the result of Flandrin et al. in some way.


Graph Implicit degree Cycles Cyclability 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bondy, J.A., Murty, U.S.R.: Graph theory with applications. Macmillan and Elsevier, London, New York (1976)CrossRefzbMATHGoogle Scholar
  2. 2.
    Broersma, H., Li, H., Li, J., Tian, F., Veldman, H.J.: Cycles through subsets with large degree sums. Discrete Math. 171, 43–54 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Dirac, G.A.: Some theorems on abstract graphs. Proc. London Math. Soc. 3, 69–81 (1952)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Flandrin, E., Li, H., Marczyk, A., Woźniak, M.: A note on a generalisation of Ore’s condition. Graphs Combin. 21, 213–216 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Fournier, I.: Thèse d’Etat. L.R.I., Université de Paris–Sud, France (1985)Google Scholar
  6. 6.
    Ore, O.: Note on Hamilton circuits. Amer. Math. Monthly 67, 55 (1960)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Ota, K.: Cycles through prescribed vertices with large degree sum. Discrete Math. 145, 201–210 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Shi, R.: 2-neighborhoods and hamiltonian condition. J. Graph Theory 16, 267–271 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Zhu, Y., Gao, J.: Implicit degrees and Chvátal’s condition for hamiltonicity. Systems Sci. Math. Sci. 2, 353–363 (1989)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Zhu, Y., Li, H., Deng, X.: Implicit-degree and circumference. Graphs Combin. 5, 283–290 (1989)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Hao Li
    • 1
    • 2
  • Wantao Ning
    • 1
  • Junqing Cai
    • 1
  1. 1.School of Mathematics and StatisticsLanzhou UniversityLanzhouChina
  2. 2.L R I, UMR 8623 CNRS and Université de Paris-Sud 11OrsayFrance

Personalised recommendations