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

Pages (from-to) | 187-205 |

Number of pages | 19 |

Journal | Graphs and Combinatorics |

Volume | 26 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2010 Mar |

Externally published | Yes |

### Keywords

- Degree sum
- k-Tree
- Specified vertices

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

*Graphs and Combinatorics*,

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

**A k-tree containing specified vertices.** / Chiba, Shuya; Matsubara, Ryota; Ozeki, Kenta; Tsugaki, Masao.

Research output: Contribution to journal › Article

*Graphs and Combinatorics*, vol. 26, no. 2, pp. 187-205. https://doi.org/10.1007/s00373-010-0903-3

}

TY - JOUR

T1 - A k-tree containing specified vertices

AU - Chiba, Shuya

AU - Matsubara, Ryota

AU - Ozeki, Kenta

AU - Tsugaki, Masao

PY - 2010/3

Y1 - 2010/3

N2 - 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.

AB - 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.

KW - Degree sum

KW - k-Tree

KW - Specified vertices

UR - http://www.scopus.com/inward/record.url?scp=77953322425&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77953322425&partnerID=8YFLogxK

U2 - 10.1007/s00373-010-0903-3

DO - 10.1007/s00373-010-0903-3

M3 - Article

VL - 26

SP - 187

EP - 205

JO - Graphs and Combinatorics

JF - Graphs and Combinatorics

SN - 0911-0119

IS - 2

ER -