### Abstract

We give sufficient conditions for a graph to have degree bounded trees. Let G be a connected graph and A a vertex subset of G. We denote by σ_{k} (A) the minimum value of the degree sum in G of any k independent vertices in A and by w (G - A) the number of components in the induced subgraph G - A. Our main results are the following: (i) If σ_{k} (A) ≥ | V (G) | - 1, then G contains a tree T with maximum degree at most k and A ⊆ V (T). (ii) If σ_{k - w (G - A)} (A) ≥ | A | - 1, then G contains a spanning tree T such that d_{T} (x) ≤ k for every x ∈ A. These are generalizations of the result by Win [S. Win, Existenz von Gerüsten mit Vorgeschriebenem Maximalgrad in Graphen, Abh. Math. Sem. Univ. Hamburg 43 (1975) 263-267] and the degree conditions are sharp.

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

Pages (from-to) | 3653-3658 |

Number of pages | 6 |

Journal | Discrete Mathematics |

Volume | 309 |

Issue number | 11 |

DOIs | |

Publication status | Published - 2009 Jun 6 |

Externally published | Yes |

### Keywords

- Degree bounded tree
- Degree sum condition
- Spanning tree
- Tree

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

*Discrete Mathematics*,

*309*(11), 3653-3658. https://doi.org/10.1016/j.disc.2007.12.099

**Degree conditions and degree bounded trees.** / Matsuda, Haruhide; Matsumura, Hajime.

Research output: Contribution to journal › Article

*Discrete Mathematics*, vol. 309, no. 11, pp. 3653-3658. https://doi.org/10.1016/j.disc.2007.12.099

}

TY - JOUR

T1 - Degree conditions and degree bounded trees

AU - Matsuda, Haruhide

AU - Matsumura, Hajime

PY - 2009/6/6

Y1 - 2009/6/6

N2 - We give sufficient conditions for a graph to have degree bounded trees. Let G be a connected graph and A a vertex subset of G. We denote by σk (A) the minimum value of the degree sum in G of any k independent vertices in A and by w (G - A) the number of components in the induced subgraph G - A. Our main results are the following: (i) If σk (A) ≥ | V (G) | - 1, then G contains a tree T with maximum degree at most k and A ⊆ V (T). (ii) If σk - w (G - A) (A) ≥ | A | - 1, then G contains a spanning tree T such that dT (x) ≤ k for every x ∈ A. These are generalizations of the result by Win [S. Win, Existenz von Gerüsten mit Vorgeschriebenem Maximalgrad in Graphen, Abh. Math. Sem. Univ. Hamburg 43 (1975) 263-267] and the degree conditions are sharp.

AB - We give sufficient conditions for a graph to have degree bounded trees. Let G be a connected graph and A a vertex subset of G. We denote by σk (A) the minimum value of the degree sum in G of any k independent vertices in A and by w (G - A) the number of components in the induced subgraph G - A. Our main results are the following: (i) If σk (A) ≥ | V (G) | - 1, then G contains a tree T with maximum degree at most k and A ⊆ V (T). (ii) If σk - w (G - A) (A) ≥ | A | - 1, then G contains a spanning tree T such that dT (x) ≤ k for every x ∈ A. These are generalizations of the result by Win [S. Win, Existenz von Gerüsten mit Vorgeschriebenem Maximalgrad in Graphen, Abh. Math. Sem. Univ. Hamburg 43 (1975) 263-267] and the degree conditions are sharp.

KW - Degree bounded tree

KW - Degree sum condition

KW - Spanning tree

KW - Tree

UR - http://www.scopus.com/inward/record.url?scp=67349259976&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=67349259976&partnerID=8YFLogxK

U2 - 10.1016/j.disc.2007.12.099

DO - 10.1016/j.disc.2007.12.099

M3 - Article

AN - SCOPUS:67349259976

VL - 309

SP - 3653

EP - 3658

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 11

ER -