### Abstract

A k-tree is a tree with maximum degree at most k. In this paper, we give sufficient conditions for a graph to have a k-tree containing specified vertices. Let k be an integer with k ≥ 3. Let G be a graph of order n and let S ⊆ V(G) with κ(S) ≥ 1. Suppose that for every l ≥ κ(S), there exists an integer t such that l≤t≤(k-1)l+2-{down left corner}_{k}/^{l-1} and the degree sum of any t independent vertices of S is at least n + tl - kl - 1. Then G has a k-tree containing S. We also show some new results on a spanning k-tree as corollaries of the above theorem.

Original language | English |
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Pages (from-to) | 187-205 |

Number of pages | 19 |

Journal | Graphs and Combinatorics |

Volume | 26 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2010 Mar 1 |

### Keywords

- Degree sum
- Specified vertices
- k-Tree

### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics

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## Cite this

Chiba, S., Matsubara, R., Ozeki, K., & Tsugaki, M. (2010). A k-tree containing specified vertices.

*Graphs and Combinatorics*,*26*(2), 187-205. https://doi.org/10.1007/s00373-010-0903-3