### Abstract

Let k and n be positive integers. Let G be a graph of order n≤4k, and let n = ∑^{k}
_{i=1} a_{i} be a partition of n into k positive integers a_{i} with 1≤a_{i}≤4. In Matsunaga (Preprint, 1995) one of the authors proved that if G is 2-connected and the minimum degree of G is at least k, then with certain types of exceptions being allowed, the vertex set of G can be decomposed into k disjoint subsets A_{l},...,A_{k} so that |A_{i}| = a_{i} and "the subgraph induced by A_{i} is connected" for all i, 1≤i≤k. In the present paper we extend this result for connected graphs in general. The paper also features the existence of D3-paths with a prescribed endvertex in a graph. (A D3-path in a graph is a path of the graph whose removal leaves no component of order at least 3.)

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

Pages (from-to) | 87-104 |

Number of pages | 18 |

Journal | Discrete Mathematics |

Volume | 213 |

Issue number | 1-3 |

Publication status | Published - 2000 Feb 28 |

### Fingerprint

### Keywords

- D-path
- Graph decomposition

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

_{3}-paths with a prescribed endvertex.

*Discrete Mathematics*,

*213*(1-3), 87-104.

**Graph decompositions and D _{3}-paths with a prescribed endvertex.** / Enomoto, Hikoe; Matsunaga, Shinsuke; Ota, Katsuhiro.

Research output: Contribution to journal › Article

_{3}-paths with a prescribed endvertex',

*Discrete Mathematics*, vol. 213, no. 1-3, pp. 87-104.

_{3}-paths with a prescribed endvertex. Discrete Mathematics. 2000 Feb 28;213(1-3):87-104.

}

TY - JOUR

T1 - Graph decompositions and D3-paths with a prescribed endvertex

AU - Enomoto, Hikoe

AU - Matsunaga, Shinsuke

AU - Ota, Katsuhiro

PY - 2000/2/28

Y1 - 2000/2/28

N2 - Let k and n be positive integers. Let G be a graph of order n≤4k, and let n = ∑k i=1 ai be a partition of n into k positive integers ai with 1≤ai≤4. In Matsunaga (Preprint, 1995) one of the authors proved that if G is 2-connected and the minimum degree of G is at least k, then with certain types of exceptions being allowed, the vertex set of G can be decomposed into k disjoint subsets Al,...,Ak so that |Ai| = ai and "the subgraph induced by Ai is connected" for all i, 1≤i≤k. In the present paper we extend this result for connected graphs in general. The paper also features the existence of D3-paths with a prescribed endvertex in a graph. (A D3-path in a graph is a path of the graph whose removal leaves no component of order at least 3.)

AB - Let k and n be positive integers. Let G be a graph of order n≤4k, and let n = ∑k i=1 ai be a partition of n into k positive integers ai with 1≤ai≤4. In Matsunaga (Preprint, 1995) one of the authors proved that if G is 2-connected and the minimum degree of G is at least k, then with certain types of exceptions being allowed, the vertex set of G can be decomposed into k disjoint subsets Al,...,Ak so that |Ai| = ai and "the subgraph induced by Ai is connected" for all i, 1≤i≤k. In the present paper we extend this result for connected graphs in general. The paper also features the existence of D3-paths with a prescribed endvertex in a graph. (A D3-path in a graph is a path of the graph whose removal leaves no component of order at least 3.)

KW - D-path

KW - Graph decomposition

UR - http://www.scopus.com/inward/record.url?scp=0043284116&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0043284116&partnerID=8YFLogxK

M3 - Article

VL - 213

SP - 87

EP - 104

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 1-3

ER -