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 |C ′∩S|>|C∩S|. We also show that if ∑4 i=1 d(a i)≥n+3+|⋂4 i=1 N(a i)| for any four independent vertices a 1, a 2, a 3, a 4 in S, then G has an S-weak-dominating S-maximum cycle C, i.e. an S-maximum cycle such that every component in G−C contains at most one vertex in S.
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Received: March 9, 1998 Revised: January 7, 1999
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Li, H. On Cycles in 3-Connected Graphs. Graphs Comb 16, 319–335 (2000). https://doi.org/10.1007/PL00007224
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DOI: https://doi.org/10.1007/PL00007224