### Abstract

In this paper, we consider forbidden subgraphs which force the existence of a 2-factor. Let G be the class of connected graphs of minimum degree at least two and maximum degree at least three, and let F _{2} be the class of graphs which have a 2-factor. For a set H of connected graphs of order at least three, a graph Gis said to be H-free if no member of H is an induced subgraph of G, and let G(H) denote the class of graphs in G that are H-free. We are interested in sets H such that G(H) is an infinite class while G(H)-F _{2} is a finite class. In particular, we investigate whether H must contain a star (i.e. K_{1,n} for some positive integer n). We prove the following. (1) If |H|=1, then H={K_{1,2}}. (2) If |H|=2, then H contains K_{1,2} or K_{1,3}. (3) If |H|=3, then H contains a star. But no restriction is imposed on the order of the star. (4) Not all of H with |H|=4 contain a star. For |H|≤2, we compare our results with a recent result by Faudree et al. (Discrete Math 308 (2008), 1571-1582), and report a difference in the conclusion when connected graphs are considered as opposed to 2-connected graphs. We also observe a phenomenon in which H does not contain a star but G(H)-G({K_{1,t}}) is finite for some t≥3.

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

Pages (from-to) | 250-266 |

Number of pages | 17 |

Journal | Journal of Graph Theory |

Volume | 64 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2010 Jul |

Externally published | Yes |

### Fingerprint

### Keywords

- 2-Factor
- Forbidden subgraph
- Induced subgraph
- Star

### ASJC Scopus subject areas

- Geometry and Topology

### Cite this

*Journal of Graph Theory*,

*64*(3), 250-266. https://doi.org/10.1002/jgt.20454

**Forbidden subgraphs and the existence of a 2-factor.** / Aldred, R. E L; Fujisawa, Jun; Saito, Akira.

Research output: Contribution to journal › Article

*Journal of Graph Theory*, vol. 64, no. 3, pp. 250-266. https://doi.org/10.1002/jgt.20454

}

TY - JOUR

T1 - Forbidden subgraphs and the existence of a 2-factor

AU - Aldred, R. E L

AU - Fujisawa, Jun

AU - Saito, Akira

PY - 2010/7

Y1 - 2010/7

N2 - In this paper, we consider forbidden subgraphs which force the existence of a 2-factor. Let G be the class of connected graphs of minimum degree at least two and maximum degree at least three, and let F 2 be the class of graphs which have a 2-factor. For a set H of connected graphs of order at least three, a graph Gis said to be H-free if no member of H is an induced subgraph of G, and let G(H) denote the class of graphs in G that are H-free. We are interested in sets H such that G(H) is an infinite class while G(H)-F 2 is a finite class. In particular, we investigate whether H must contain a star (i.e. K1,n for some positive integer n). We prove the following. (1) If |H|=1, then H={K1,2}. (2) If |H|=2, then H contains K1,2 or K1,3. (3) If |H|=3, then H contains a star. But no restriction is imposed on the order of the star. (4) Not all of H with |H|=4 contain a star. For |H|≤2, we compare our results with a recent result by Faudree et al. (Discrete Math 308 (2008), 1571-1582), and report a difference in the conclusion when connected graphs are considered as opposed to 2-connected graphs. We also observe a phenomenon in which H does not contain a star but G(H)-G({K1,t}) is finite for some t≥3.

AB - In this paper, we consider forbidden subgraphs which force the existence of a 2-factor. Let G be the class of connected graphs of minimum degree at least two and maximum degree at least three, and let F 2 be the class of graphs which have a 2-factor. For a set H of connected graphs of order at least three, a graph Gis said to be H-free if no member of H is an induced subgraph of G, and let G(H) denote the class of graphs in G that are H-free. We are interested in sets H such that G(H) is an infinite class while G(H)-F 2 is a finite class. In particular, we investigate whether H must contain a star (i.e. K1,n for some positive integer n). We prove the following. (1) If |H|=1, then H={K1,2}. (2) If |H|=2, then H contains K1,2 or K1,3. (3) If |H|=3, then H contains a star. But no restriction is imposed on the order of the star. (4) Not all of H with |H|=4 contain a star. For |H|≤2, we compare our results with a recent result by Faudree et al. (Discrete Math 308 (2008), 1571-1582), and report a difference in the conclusion when connected graphs are considered as opposed to 2-connected graphs. We also observe a phenomenon in which H does not contain a star but G(H)-G({K1,t}) is finite for some t≥3.

KW - 2-Factor

KW - Forbidden subgraph

KW - Induced subgraph

KW - Star

UR - http://www.scopus.com/inward/record.url?scp=77954343695&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77954343695&partnerID=8YFLogxK

U2 - 10.1002/jgt.20454

DO - 10.1002/jgt.20454

M3 - Article

AN - SCOPUS:77954343695

VL - 64

SP - 250

EP - 266

JO - Journal of Graph Theory

JF - Journal of Graph Theory

SN - 0364-9024

IS - 3

ER -