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Graphs and Combinatorics

, Volume 29, Issue 5, pp 1459–1469 | Cite as

An Implicit Degree Condition for Pancyclicity of Graphs

  • Hao LiEmail author
  • Junqing Cai
  • Wantao Ning
Original Paper

Abstract

Zhu, Li and Deng introduced in 1989 the definition of implicit degree of a vertex v in a graph G, denoted by id(v), by using the degrees of the vertices in its neighborhood and the second neighborhood. And they obtained sufficient conditions with implicit degrees for a graph to be hamiltonian. In this paper, we prove that if G is a 2–connected graph of order n ≥ 3 such that id(v) ≥ n/2 for each vertex v of G, then G is pancyclic unless G is bipartite, or else n = 4r, r ≥ 2 and G is in a class of graphs F 4r defined in the paper.

Keywords

Implicit degree Pancyclic Hamiltonian cycles Graph 

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References

  1. 1.
    Benhocine A., Wojda A.P.: The Geng-Hua Fan conditions for pancyclic or Hamilton-connected graphs. J. Comb. Theory Ser. B 42, 167–180 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Bondy J.A.: Pancyclic graphs, I. J. Comb. Theory Ser. B 11, 80–84 (1971)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Bondy J.A., Murty U.S.R.: Graph Theory with Applications. Macmillan Press, London (1976)zbMATHGoogle Scholar
  4. 5.
    Chen G.: Longest cycles in 2-connected graphs. J. Central China Normal Univ. Natur. Sci. 3, 39–42 (1986)Google Scholar
  5. 5.
    Chen B., Chen B.: An implicit degree condition for long cycles in 2-connected graphs. Appl. Math. Lett. 19, 1148–1151 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Dirac G.A.: Some theorems on abstract graphs. Proc. Lond. Math. Soc. 2, 69–81 (1952)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Ore O.: Note on hamilton circuits. Am. Math. Mon. 67, 55 (1960)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Schmeichel E., Hakimi S.: Pancyclic graphs and a conjecture of Bondy and Chavátal. J. Comb. Theory Ser. B 17, 23–34 (1974)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Schmeichel E., Hakimi S.: A cycle structure theo for hamiltonian graphs. J. Comb. Theory Ser. B 45, 99–107 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Zhu Y., Li H., Deng X.: Implicit-degrees and circumferences. Graphs Comb. 5, 283–290 (1989)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer 2012

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsLanzhou UniversityLanzhouPeople’s Republic of China
  2. 2.L.R.I., UMR 8623, CNRS and Université Paris-Sud 11OrsayFrance
  3. 3.Department of MathematicsXidian UniversityXianPeople’s Republic of China

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