### Abstract

A k-tree of a graph is a spanning tree with maximum degree at most k. We give sufficient conditions for a graph G to have a k-tree with specified leaves: Let k,s, and n be integers such that k ≥ 2, 0 ≤ s ≤ k, and n ≤ s+1. Suppose that (1) G is (s+1)-connected and the degree sum of any k independent vertices of G is at least |G|+(k-1)s-1, or (2) G is n-connected and the independence number of G is at most (n-s)(k-1)+1. Then for any s specified vertices of G, G has a k-tree containing them as leaves. We also discuss the sharpness of the results.

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

Pages (from-to) | 371-381 |

Number of pages | 11 |

Journal | Graphs and Combinatorics |

Volume | 22 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2006 Nov |

### Keywords

- Factor
- Independence number
- Ore
- Spanning subgraph
- Tree

### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics

## Fingerprint Dive into the research topics of 'On a k-tree containing specified leaves in a graph'. Together they form a unique fingerprint.

## Cite this

Matsuda, H., & Matsumura, H. (2006). On a k-tree containing specified leaves in a graph.

*Graphs and Combinatorics*,*22*(3), 371-381. https://doi.org/10.1007/s00373-006-0660-5