Graphs and Combinatorics

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

On Cycles in 3-Connected Graphs


 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.


Minimum Degree Independent Vertex 
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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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