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


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.


Implicit degree Pancyclic Hamiltonian cycles Graph 


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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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