DMGT

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

THE CHVÁTAL-ERDOS CONDITION AND 2-FACTORS WITH A SPECIFIED NUMBER OF COMPONENTS

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

Abstract

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.

References

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