### Abstract

In [3], Faudree et al. considered the proposition “Every (X, Y)-free graph of sufficiently large order has a 2-factor,” and they determined those pairs (X, Y) which make this proposition true. Their result says that one of them is (X, Y) = (K_{1,4}, P_{4}). In this paper, we investigate the existence of 2-factors in r-connected (K_{1, k}, P_{4})-free graphs. We prove that if r ≥ 1 and k ≥ 2, and if G is an r-connected (K_{1, k}, P_{4})-free graph with minimum degree at least k − 1, then G has a 2-factor with at most max(k − r, 1) components unless (k − 1)K_{2} + (k − 2)K_{1} ⊆ G ⊆ (k − 1)K_{2} + K_{k−2}. The bound on the minimum degree is best possible.

Original language | English |
---|---|

Pages (from-to) | 415-420 |

Number of pages | 6 |

Journal | Tokyo Journal of Mathematics |

Volume | 31 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2008 Jan 1 |

### Fingerprint

### Keywords

- 2-factor
- Forbidden subgraph
- Minimum degree

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

_{1, k}, P

_{4})-Free Graphs.

*Tokyo Journal of Mathematics*,

*31*(2), 415-420. https://doi.org/10.3836/tjm/1233844061

**On 2-Factors in r-Connected (K _{1, k}, P_{4})-Free Graphs.** / Egawa, Yoshimi; Fujisawa, Jun; Fujita, Shinya; Ota, Katsuhiro.

Research output: Contribution to journal › Article

_{1, k}, P

_{4})-Free Graphs',

*Tokyo Journal of Mathematics*, vol. 31, no. 2, pp. 415-420. https://doi.org/10.3836/tjm/1233844061

_{1, k}, P

_{4})-Free Graphs. Tokyo Journal of Mathematics. 2008 Jan 1;31(2):415-420. https://doi.org/10.3836/tjm/1233844061

}

TY - JOUR

T1 - On 2-Factors in r-Connected (K1, k, P4)-Free Graphs

AU - Egawa, Yoshimi

AU - Fujisawa, Jun

AU - Fujita, Shinya

AU - Ota, Katsuhiro

PY - 2008/1/1

Y1 - 2008/1/1

N2 - In [3], Faudree et al. considered the proposition “Every (X, Y)-free graph of sufficiently large order has a 2-factor,” and they determined those pairs (X, Y) which make this proposition true. Their result says that one of them is (X, Y) = (K1,4, P4). In this paper, we investigate the existence of 2-factors in r-connected (K1, k, P4)-free graphs. We prove that if r ≥ 1 and k ≥ 2, and if G is an r-connected (K1, k, P4)-free graph with minimum degree at least k − 1, then G has a 2-factor with at most max(k − r, 1) components unless (k − 1)K2 + (k − 2)K1 ⊆ G ⊆ (k − 1)K2 + Kk−2. The bound on the minimum degree is best possible.

AB - In [3], Faudree et al. considered the proposition “Every (X, Y)-free graph of sufficiently large order has a 2-factor,” and they determined those pairs (X, Y) which make this proposition true. Their result says that one of them is (X, Y) = (K1,4, P4). In this paper, we investigate the existence of 2-factors in r-connected (K1, k, P4)-free graphs. We prove that if r ≥ 1 and k ≥ 2, and if G is an r-connected (K1, k, P4)-free graph with minimum degree at least k − 1, then G has a 2-factor with at most max(k − r, 1) components unless (k − 1)K2 + (k − 2)K1 ⊆ G ⊆ (k − 1)K2 + Kk−2. The bound on the minimum degree is best possible.

KW - 2-factor

KW - Forbidden subgraph

KW - Minimum degree

UR - http://www.scopus.com/inward/record.url?scp=84936756124&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84936756124&partnerID=8YFLogxK

U2 - 10.3836/tjm/1233844061

DO - 10.3836/tjm/1233844061

M3 - Article

AN - SCOPUS:84936756124

VL - 31

SP - 415

EP - 420

JO - Tokyo Journal of Mathematics

JF - Tokyo Journal of Mathematics

SN - 0387-3870

IS - 2

ER -