### 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 |
---|---|

Pages (from-to) | 429-437 |

Number of pages | 9 |

Journal | Graphs and Combinatorics |

Volume | 30 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2014 Mar |

### Keywords

- Branch vertices
- Claw-free graphs
- Spanning trees

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

*Graphs and Combinatorics*,

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

**Spanning Trees with a Bounded Number of Branch Vertices in a Claw-Free Graph.** / Matsuda, Haruhide; Ozeki, Kenta; Yamashita, Tomoki.

Research output: Contribution to journal › Article

*Graphs and Combinatorics*, vol. 30, no. 2, pp. 429-437. https://doi.org/10.1007/s00373-012-1277-5

}

TY - JOUR

T1 - Spanning Trees with a Bounded Number of Branch Vertices in a Claw-Free Graph

AU - Matsuda, Haruhide

AU - Ozeki, Kenta

AU - Yamashita, Tomoki

PY - 2014/3

Y1 - 2014/3

N2 - 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.

AB - 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.

KW - Branch vertices

KW - Claw-free graphs

KW - Spanning trees

UR - http://www.scopus.com/inward/record.url?scp=84894300730&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84894300730&partnerID=8YFLogxK

U2 - 10.1007/s00373-012-1277-5

DO - 10.1007/s00373-012-1277-5

M3 - Article

AN - SCOPUS:84894300730

VL - 30

SP - 429

EP - 437

JO - Graphs and Combinatorics

JF - Graphs and Combinatorics

SN - 0911-0119

IS - 2

ER -