### Abstract

A graph is called a weighted graph when each edge e is assigned a nonnegative number w(e), called the weight of e. For a vertex v of a weighted graph, d ^{w} (v) is the sum of the weights of the edges incident with v. For a subgraph H of a weighted graph G, the weight of H is the sum of the weights of the edges belonging to H. In this paper, we give a new sufficient condition for a weighted graph to have a heavy cycle. A 2-connected weighted graph G contains either a Hamilton cycle or a cycle of weight at least c, if G satisfies the following conditions: In every induced claw or induced modified claw F of G, (1) max{d ^{w} (x),d ^{w} (y)}≤ c/2 for each non-adjacent pair of vertices x and y in F, and (2) all edges of F have the same weight.

Original language | English |
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Pages (from-to) | 217-229 |

Number of pages | 13 |

Journal | Graphs and Combinatorics |

Volume | 21 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2005 Jun |

### Fingerprint

### Keywords

- Claw
- Fan-type condition
- Heavy cycle
- Modified claw
- Weighted graph

### ASJC Scopus subject areas

- Mathematics(all)
- Discrete Mathematics and Combinatorics

### Cite this

**Claw conditions for heavy cycles in weighted graphs.** / Fujisawa, Jun.

Research output: Contribution to journal › Article

*Graphs and Combinatorics*, vol. 21, no. 2, pp. 217-229. https://doi.org/10.1007/s00373-005-0607-2

}

TY - JOUR

T1 - Claw conditions for heavy cycles in weighted graphs

AU - Fujisawa, Jun

PY - 2005/6

Y1 - 2005/6

N2 - A graph is called a weighted graph when each edge e is assigned a nonnegative number w(e), called the weight of e. For a vertex v of a weighted graph, d w (v) is the sum of the weights of the edges incident with v. For a subgraph H of a weighted graph G, the weight of H is the sum of the weights of the edges belonging to H. In this paper, we give a new sufficient condition for a weighted graph to have a heavy cycle. A 2-connected weighted graph G contains either a Hamilton cycle or a cycle of weight at least c, if G satisfies the following conditions: In every induced claw or induced modified claw F of G, (1) max{d w (x),d w (y)}≤ c/2 for each non-adjacent pair of vertices x and y in F, and (2) all edges of F have the same weight.

AB - A graph is called a weighted graph when each edge e is assigned a nonnegative number w(e), called the weight of e. For a vertex v of a weighted graph, d w (v) is the sum of the weights of the edges incident with v. For a subgraph H of a weighted graph G, the weight of H is the sum of the weights of the edges belonging to H. In this paper, we give a new sufficient condition for a weighted graph to have a heavy cycle. A 2-connected weighted graph G contains either a Hamilton cycle or a cycle of weight at least c, if G satisfies the following conditions: In every induced claw or induced modified claw F of G, (1) max{d w (x),d w (y)}≤ c/2 for each non-adjacent pair of vertices x and y in F, and (2) all edges of F have the same weight.

KW - Claw

KW - Fan-type condition

KW - Heavy cycle

KW - Modified claw

KW - Weighted graph

UR - http://www.scopus.com/inward/record.url?scp=21544444165&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=21544444165&partnerID=8YFLogxK

U2 - 10.1007/s00373-005-0607-2

DO - 10.1007/s00373-005-0607-2

M3 - Article

AN - SCOPUS:21544444165

VL - 21

SP - 217

EP - 229

JO - Graphs and Combinatorics

JF - Graphs and Combinatorics

SN - 0911-0119

IS - 2

ER -