### 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 |

Externally published | Yes |

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)
- Discrete Mathematics and Combinatorics

### Cite this

*Graphs and Combinatorics*,

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

**On a k-tree containing specified leaves in a graph.** / Matsuda, Haruhide; Matsumura, Hajime.

Research output: Contribution to journal › Article

*Graphs and Combinatorics*, vol. 22, no. 3, pp. 371-381. https://doi.org/10.1007/s00373-006-0660-5

}

TY - JOUR

T1 - On a k-tree containing specified leaves in a graph

AU - Matsuda, Haruhide

AU - Matsumura, Hajime

PY - 2006/11

Y1 - 2006/11

N2 - 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.

AB - 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.

KW - Factor

KW - Independence number

KW - Ore

KW - Spanning subgraph

KW - Tree

UR - http://www.scopus.com/inward/record.url?scp=33751548502&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33751548502&partnerID=8YFLogxK

U2 - 10.1007/s00373-006-0660-5

DO - 10.1007/s00373-006-0660-5

M3 - Article

AN - SCOPUS:33751548502

VL - 22

SP - 371

EP - 381

JO - Graphs and Combinatorics

JF - Graphs and Combinatorics

SN - 0911-0119

IS - 3

ER -