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An Implicit Degree Condition for Pancyclicity of Graphs

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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.

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Author notes

  1. Hao Li

    Present address: L.R.I., UMR 8623, CNRS and Université Paris-Sud 11, 91405, Orsay, France

Authors and Affiliations

  1. School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, People’s Republic of China

    Hao Li & Junqing Cai

  2. Department of Mathematics, Xidian University, Xian, Shaanxi, 710071, People’s Republic of China

    Wantao Ning

Authors
  1. Hao Li
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  2. Junqing Cai
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  3. Wantao Ning
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Correspondence to Hao Li.

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Li, H., Cai, J. & Ning, W. An Implicit Degree Condition for Pancyclicity of Graphs. Graphs and Combinatorics 29, 1459–1469 (2013). https://doi.org/10.1007/s00373-012-1198-3

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  • DOI: https://doi.org/10.1007/s00373-012-1198-3

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