Discussiones Mathematicae Graph Theory 27(3) (2007) 401-407
doi: 10.7151/dmgt.1370

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Guantao Chen

Department of Mathematics and Statistics
Georgia State University, Atlanta, GA 30303, USA

Ronald J. Gould

Department of Mathematics and Computer Science
Emory University, Atlanta, GA 30322, USA

Ken-ichi Kawarabayashi

National Institute of Informatics
2-1-2 Hitotsubashi, Chiyoda-Ku, Tokyo 101-8430, Japan

Katsuhiro Ota

Department of Mathematics, Keio University
3-14-1 Hiyoshi, Kohoku-Ku, Yokohama 223-8522, Japan

Akira Saito

Department of Computer Science, Nihon University
Sakurajosui 3-25-40, Setagaya-Ku, Tokyo 156-8550, Japan

Ingo Schiermeyer

Institut für Diskrete Mathematik und Algebra
Technische Universität
Bergakademie Freiberg, D-09596 Freiberg, Germany


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.

The Chvátal-Erdös Condition and 2-Factors with ...

Keywords: Chvátal-Erdös condition, 2-factor, hamiltonian cycle, Ramsey number.

2000 Mathematics Subject Classification: Primary: 05C38; Secondary: 05C40, 05C45, 05C69.


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Received 1 March 2006
Revised 23 July 2007
Accepted 23 July 2007