### Abstract

A weighted graph is one in which every edge e is assigned a nonnegative number w(e), called the weight of e. For a vertex v of a weighted graph, dw(v) is the sum of the weights of the edges incident to v. And the weight of a path is the sum of the weights of the edges belonging to it. In this paper, we give a sufficient condition for a weighted graph to have a heavy path which joins two specified vertices. Let G be a 2-connected weighted graph and let x and y be distinct vertices of G. Suppose that dw(u)+dw(v)≥2d for every pair of non-adjacent vertices u and v∈V(G)\{x,y}. Then x and y are joined by a path of weight at least d, or they are joined by a Hamilton path. Also, we consider the case when G has some vertices whose weighted degree are not assumed.

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

Pages (from-to) | 100-109 |

Number of pages | 10 |

Journal | Discrete Mathematics |

Volume | 300 |

Issue number | 1-3 |

DOIs | |

Publication status | Published - 2005 Sep 6 |

### Fingerprint

### Keywords

- Heavy path
- Ore-type degree condition
- Weighted graph

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

*Discrete Mathematics*,

*300*(1-3), 100-109. https://doi.org/10.1016/j.disc.2005.06.006

**Ore-type degree condition for heavy paths in weighted graphs.** / Enomoto, Hikoe; Fujisawa, Jun; Ota, Katsuhiro.

Research output: Contribution to journal › Article

*Discrete Mathematics*, vol. 300, no. 1-3, pp. 100-109. https://doi.org/10.1016/j.disc.2005.06.006

}

TY - JOUR

T1 - Ore-type degree condition for heavy paths in weighted graphs

AU - Enomoto, Hikoe

AU - Fujisawa, Jun

AU - Ota, Katsuhiro

PY - 2005/9/6

Y1 - 2005/9/6

N2 - A weighted graph is one in which every edge e is assigned a nonnegative number w(e), called the weight of e. For a vertex v of a weighted graph, dw(v) is the sum of the weights of the edges incident to v. And the weight of a path is the sum of the weights of the edges belonging to it. In this paper, we give a sufficient condition for a weighted graph to have a heavy path which joins two specified vertices. Let G be a 2-connected weighted graph and let x and y be distinct vertices of G. Suppose that dw(u)+dw(v)≥2d for every pair of non-adjacent vertices u and v∈V(G)\{x,y}. Then x and y are joined by a path of weight at least d, or they are joined by a Hamilton path. Also, we consider the case when G has some vertices whose weighted degree are not assumed.

AB - A weighted graph is one in which every edge e is assigned a nonnegative number w(e), called the weight of e. For a vertex v of a weighted graph, dw(v) is the sum of the weights of the edges incident to v. And the weight of a path is the sum of the weights of the edges belonging to it. In this paper, we give a sufficient condition for a weighted graph to have a heavy path which joins two specified vertices. Let G be a 2-connected weighted graph and let x and y be distinct vertices of G. Suppose that dw(u)+dw(v)≥2d for every pair of non-adjacent vertices u and v∈V(G)\{x,y}. Then x and y are joined by a path of weight at least d, or they are joined by a Hamilton path. Also, we consider the case when G has some vertices whose weighted degree are not assumed.

KW - Heavy path

KW - Ore-type degree condition

KW - Weighted graph

UR - http://www.scopus.com/inward/record.url?scp=25444499699&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=25444499699&partnerID=8YFLogxK

U2 - 10.1016/j.disc.2005.06.006

DO - 10.1016/j.disc.2005.06.006

M3 - Article

AN - SCOPUS:25444499699

VL - 300

SP - 100

EP - 109

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 1-3

ER -