Geodesic
The Chvátal-Erdős condition and 2-factors with a specified number of components
Discussiones Mathematicae. Graph Theory, Tome 27 (2007) no. 3, pp. 401-407.

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Let G be a 2-connected graph of order n satisfying α(G) = a ≤ κ(G), where α(G) and κ(G) are the independence number and the connectivity of G, respectively, and let r(m,n) denote the Ramsey number. The well-known Chvátal-Erdös Theorem states that G has a hamiltonian cycle. In this paper, we extend this theorem, and prove that G has a 2-factor with a specified number of components if n is sufficiently large. More precisely, we prove that (1) if n ≥ k·r(a+4, a+1), then G has a 2-factor with k components, and (2) if n ≥ r(2a+3, a+1)+3(k-1), then G has a 2-factor with k components such that all components but one have order three.
Keywords: Chvátal-Erdös condition, 2-factor, hamiltonian cycle, Ramsey number 
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Chen, Guantao; Gould, Ronald; Kawarabayashi, Ken-ichi; Ota, Katsuhiro; Saito, Akira; Schiermeyer, Ingo. The Chvátal-Erdős condition and 2-factors with a specified number of components. Discussiones Mathematicae. Graph Theory, Tome 27 (2007) no. 3, pp. 401-407. http://geodesic.mathdoc.fr/item/DMGT_2007_27_3_a1/

[1] A. Bondy, A remark on two sufficient conditions for Hamilton cycles, Discrete Math. 22 (1978) 191-194, doi: 10.1016/0012-365X(78)90124-3.

[2] S. Brandt, G. Chen, R. Faudree, R. Gould and L. Lesniak, Degree conditions for 2-factors, J. Graph Theory 24 (1997) 165-173, doi: 10.1002/(SICI)1097-0118(199702)24:2165::AID-JGT4>3.0.CO;2-O

[3] G. Chartrand and L. Lesniak, Graphs Digraphs (3rd ed.) (Wadsworth Brooks/Cole, Monterey, CA, 1996).

[4] V. Chvátal and P. Erdös, A note on hamiltonian circuits, Discrete Math. 2 (1972) 111-113, doi: 10.1016/0012-365X(72)90079-9.

[5] Y. Egawa, personal communication.

[6] H. Enomoto, On the existence of disjoint cycles in a graph, Combinatorica 18 (1998) 487-492, doi: 10.1007/s004930050034.

[7] A. Kaneko and K. Yoshimoto, A 2-factor with two components of a graph satisfying the Chvátal-Erdös condition, J. Graph Theory 43 (2003) 269-279, doi: 10.1002/jgt.10119.

[8] O. Ore, Note on Hamilton circuits, Amer. Math. Monthly 67 (1960) 55, doi: 10.2307/2308928.