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Graphs and Combinatorics

, Volume 16, Issue 3, pp 319–335 | Cite as

On Cycles in 3-Connected Graphs

Abstract.

 Let G be a 3-connected graph of order n and S a subset of vertices. Denote by δ(S) the minimum degree (in G) of vertices of S. Then we prove that the circumference of G is at least min{|S|, 2δ(S)} if the degree sum of any four independent vertices of S is at least n+6. A cycle C is called S-maximum if there is no cycle C with |CS|>|CS|. We also show that if ∑4i=1d(ai)≥n+3+|⋂4i=1N(ai)| for any four independent vertices a1, a2, a3, a4 in S, then G has an S-weak-dominating S-maximum cycle C, i.e. an S-maximum cycle such that every component in GC contains at most one vertex in S.

Keywords

Minimum Degree Independent Vertex 
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.

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Copyright information

© Springer-Verlag Tokyo 2000

Authors and Affiliations

  • Hao Li
    • 1
  1. 1.Laboratoire de Recherche en Informatique, URA 410, C.N.R.S. Bât. 490, Université de Paris-sud, 91405-Orsay cedex, France. e-mail: li@lri.frFR

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