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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     author = {Chen, Guantao and Gould, Ronald and Kawarabayashi, Ken-ichi and Ota, Katsuhiro and Saito, Akira and Schiermeyer, Ingo},
     title = {The {Chv\'atal-Erd\H{o}s} condition and 2-factors with a specified number of components},
     journal = {Discussiones Mathematicae. Graph Theory},
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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/