# The Chvátal-Erdős condition and 2-factors with a specified number of components

Guantao Chen; Ronald J. Gould; Ken-ichi Kawarabayashi; Katsuhiro Ota; Akira Saito; Ingo Schiermeyer

Discussiones Mathematicae Graph Theory (2007)

- Volume: 27, Issue: 3, page 401-407
- ISSN: 2083-5892

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topGuantao Chen, et al. "The Chvátal-Erdős condition and 2-factors with a specified number of components." Discussiones Mathematicae Graph Theory 27.3 (2007): 401-407. <http://eudml.org/doc/270518>.

@article{GuantaoChen2007,

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.},

author = {Guantao Chen, Ronald J. Gould, Ken-ichi Kawarabayashi, Katsuhiro Ota, Akira Saito, Ingo Schiermeyer},

journal = {Discussiones Mathematicae Graph Theory},

keywords = {Chvátal-Erdös condition; 2-factor; hamiltonian cycle; Ramsey number; Chvátal-Erdős condition},

language = {eng},

number = {3},

pages = {401-407},

title = {The Chvátal-Erdős condition and 2-factors with a specified number of components},

url = {http://eudml.org/doc/270518},

volume = {27},

year = {2007},

}

TY - JOUR

AU - Guantao Chen

AU - Ronald J. Gould

AU - Ken-ichi Kawarabayashi

AU - Katsuhiro Ota

AU - Akira Saito

AU - Ingo Schiermeyer

TI - The Chvátal-Erdős condition and 2-factors with a specified number of components

JO - Discussiones Mathematicae Graph Theory

PY - 2007

VL - 27

IS - 3

SP - 401

EP - 407

AB - 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.

LA - eng

KW - Chvátal-Erdös condition; 2-factor; hamiltonian cycle; Ramsey number; Chvátal-Erdős condition

UR - http://eudml.org/doc/270518

ER -

## References

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- [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. Zbl1033.05085
- [8] O. Ore, Note on Hamilton circuits, Amer. Math. Monthly 67 (1960) 55, doi: 10.2307/2308928.

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