### Abstract

We consider the existence of several different kinds of factors in 4-connected claw-free graphs. This is motivated by the following two conjectures which are in fact equivalent by a recent result of the third author. Conjecture 1 (Thomassen): Every 4-connected line graph is hamiltonian, i.e., has a connected 2-factor. Conjecture 2 (Matthews and Sumner): Every 4-connected claw-free graph is hamiltonian. We first show that Conjecture 2 is true within the class of hourglass-free graphs, i.e., graphs that do not contain an induced subgraph isomorphic to two triangles meeting in exactly one vertex. Next we show that a weaker form of Conjecture 2 is true, in which the conclusion is replaced by the conclusion that there exists a connected spanning subgraph in which each vertex has degree two or four. Finally we show that Conjectures 1 and 2 are equivalent to seemingly weaker conjectures in which the conclusion is replaced by the conclusion that there exists a spanning subgraph consisting of a bounded number of paths.

Original language | Undefined |
---|---|

Pages (from-to) | 125-136 |

Number of pages | 12 |

Journal | Journal of graph theory |

Volume | 37 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2001 |

### Keywords

- METIS-201534
- Line graph
- Factor
- (Hamilton) cycle
- Claw-free graph
- IR-71757
- Hamilton path

## Cite this

Broersma, H. J., Kriesell, M., Kriesell, M., & Ryjacek, Z. (2001). On factors of 4-connected claw-free graphs.

*Journal of graph theory*,*37*(2), 125-136. https://doi.org/10.1002/jgt.1008