### Abstract

In this article, we study the existence of a 2-factor in a K _{1, n}-free graph. Sumner [J London Math Soc 13 (1976), 351-359] proved that for n ≥ 4, an (n-1)-connected K_{1, n}-free graph of even order has a 1-factor. On the other hand, for every pair of integers m and n with m ≥ n ≥ 4, there exist infinitely many (n-2)-connected K_{1, n}-free graphs of even order and minimum degree at least m which have no 1-factor. This implies that the connectivity condition of Sumner's result is sharp, and we cannot guarantee the existence of a 1-factor by imposing a large minimum degree. On the other hand, Ota and Tokuda [J Graph Theory 22 (1996), 59-64] proved that for n ≥ 3, every K_{1, n}-free graph of minimum degree at least 2n-2 has a 2-factor, regardless of its connectivity. They also gave examples showing that their minimum degree condition is sharp. But all of them have bridges. These suggest that the effects of connectivity, edge-connectivity and minimum degree to the existence of a 2-factor in a K_{1, n}-free graph are more complicated than those to the existence of a 1-factor. In this article, we clarify these effects by giving sharp minimum degree conditions for a K _{1, n}-free graph with a given connectivity or edge-connectivity to have a 2-factor.

Original language | English |
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Pages (from-to) | 77-89 |

Number of pages | 13 |

Journal | Journal of Graph Theory |

Volume | 68 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2011 Sep 1 |

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### Keywords

- (edge-)connectivity
- 2-factor
- K-free graph
- minimum degree

### ASJC Scopus subject areas

- Geometry and Topology

### Cite this

_{1, n}-free graphs with large connectivity and large edge-connectivity.

*Journal of Graph Theory*,

*68*(1), 77-89. https://doi.org/10.1002/jgt.20541