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• CNRS Université de Parissud XI

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• Theorem:

cyc(G) ≥9 for every 3connected cubic graph G. This bound is sharp (The Petersen graph). ('4:;:)""!#" (2"/2! "<,-=<1/20< Theorem: If G is 3connected and planar, cyc(G) ≥ 23. This bound is sharp. >%(!!4('!4:; <//2! $",1,---/./2/.0

• Theorem: Let G be a 3connected clawfree graph and let

U= {u1,...,uk }, k≤ ≤ ≤ ≤ 9, be an arbitrary set of at most nine vertices in G. Then G contains a cycle C which contains U.

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Theorem: If G is 3connected and clawfree, then cyc(G)≥ 6.

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New result

Theorem (Flandrin, Gyori, Li, Shu, ): Let be a free graph and a connected subset of vertices in . Then if ≤ ≤ ≤ ≤ , there exists a cycle containing .

Corollary. If G is free kconnected, then the cyclability cyc(G) is at least .

New result

Theorem (Zhang, Li): Let be a 3edgeconnected graph and a weakkedge connected vertex subset of vertices in ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ . Then G admits an eulerian subgraph containing all vertices of S A vertex set S ⊆ ⊆ ⊆ ⊆ V(G) is weakkedgeconnected if for every subset C of S and x ∈ SC, there are min{k,|C|} edgedisjoint (x,C)paths in G. The condition “ 3edgeconnected” is necessary:

G=K2,2m+1and S is the (2m+1) part.

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谢谢 谢谢 谢谢 谢谢%;FC?$@

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