### Abstract

Let k be a non-negative integer. A branch vertex of a tree is a vertex of degree at least three. We show two sufficient conditions for a connected claw-free graph to have a spanning tree with a bounded number of branch vertices: (i) A connected claw-free graph has a spanning tree with at most k branch vertices if its independence number is at most 2k + 2. (ii) A connected claw-free graph of order n has a spanning tree with at most one branch vertex if the degree sum of any five independent vertices is at least n - 2. These conditions are best possible. A related conjecture also is proposed.

Original language | English |
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Pages (from-to) | 429-437 |

Number of pages | 9 |

Journal | Graphs and Combinatorics |

Volume | 30 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2014 Mar 1 |

### Keywords

- Branch vertices
- Claw-free graphs
- Spanning trees

### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics

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## Cite this

Matsuda, H., Ozeki, K., & Yamashita, T. (2014). Spanning Trees with a Bounded Number of Branch Vertices in a Claw-Free Graph.

*Graphs and Combinatorics*,*30*(2), 429-437. https://doi.org/10.1007/s00373-012-1277-5