Geodesic
2-factors in claw-free graphs
Discussiones Mathematicae. Graph Theory, Tome 20 (2000) no. 2, pp. 165-172

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We consider the question of the range of the number of cycles possible in a 2-factor of a 2-connected claw-free graph with sufficiently high minimum degree. (By claw-free we mean the graph has no induced K_1,3.) In particular, we show that for such a graph G of order n ≥ 51 with δ(G) ≥ (n-2)/3, G contains a 2-factor with exactly k cycles, for 1 ≤ k ≤ (n-24)/3. We also show that this result is sharp in the sense that if we lower δ(G), we cannot obtain the full range of values for k.
Keywords: claw-free, forbidden subgraphs, 2-factors, cycles 
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Chen, Guantao; Faudree, Jill; Gould, Ronald; Saito, Akira. 2-factors in claw-free graphs. Discussiones Mathematicae. Graph Theory, Tome 20 (2000) no. 2, pp. 165-172. http://geodesic.mathdoc.fr/item/DMGT_2000_20_2_a0/