### Abstract

A set of paths joining a vertex y and a vertex set L is called (y, L)-fan if any two of the paths have only y in common, and its width is the number of paths forming it. In weighted graphs, it is known that the existence of heavy fan is useful to find a heavy cycle containing some specified vertices. In this paper, we show the existence of heavy fans with large width containing some specified vertices in weighted graphs of large connectivity, which is a weighted analogue of Perfect's theorem. Using this, in 3-connected weighted graphs, we can find heavy cycles containing three specified vertices, and also heavy paths joining two specified vertices containing two more specified vertices. These results extend the previous results in 2-connected weighted graphs to 3-connected weighted graphs.

Original language | English |
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Pages (from-to) | 38-53 |

Number of pages | 16 |

Journal | Discrete Mathematics |

Volume | 307 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2007 Jan 6 |

### Fingerprint

### Keywords

- Heavy cycle
- Heavy path
- Perfect's theorem
- Specified vertex
- Weighted graph

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

**Heavy fans, cycles and paths in weighted graphs of large connectivity.** / Fujisawa, Jun.

Research output: Contribution to journal › Article

*Discrete Mathematics*, vol. 307, no. 1, pp. 38-53. https://doi.org/10.1016/j.disc.2006.04.036

}

TY - JOUR

T1 - Heavy fans, cycles and paths in weighted graphs of large connectivity

AU - Fujisawa, Jun

PY - 2007/1/6

Y1 - 2007/1/6

N2 - A set of paths joining a vertex y and a vertex set L is called (y, L)-fan if any two of the paths have only y in common, and its width is the number of paths forming it. In weighted graphs, it is known that the existence of heavy fan is useful to find a heavy cycle containing some specified vertices. In this paper, we show the existence of heavy fans with large width containing some specified vertices in weighted graphs of large connectivity, which is a weighted analogue of Perfect's theorem. Using this, in 3-connected weighted graphs, we can find heavy cycles containing three specified vertices, and also heavy paths joining two specified vertices containing two more specified vertices. These results extend the previous results in 2-connected weighted graphs to 3-connected weighted graphs.

AB - A set of paths joining a vertex y and a vertex set L is called (y, L)-fan if any two of the paths have only y in common, and its width is the number of paths forming it. In weighted graphs, it is known that the existence of heavy fan is useful to find a heavy cycle containing some specified vertices. In this paper, we show the existence of heavy fans with large width containing some specified vertices in weighted graphs of large connectivity, which is a weighted analogue of Perfect's theorem. Using this, in 3-connected weighted graphs, we can find heavy cycles containing three specified vertices, and also heavy paths joining two specified vertices containing two more specified vertices. These results extend the previous results in 2-connected weighted graphs to 3-connected weighted graphs.

KW - Heavy cycle

KW - Heavy path

KW - Perfect's theorem

KW - Specified vertex

KW - Weighted graph

UR - http://www.scopus.com/inward/record.url?scp=33751209482&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33751209482&partnerID=8YFLogxK

U2 - 10.1016/j.disc.2006.04.036

DO - 10.1016/j.disc.2006.04.036

M3 - Article

VL - 307

SP - 38

EP - 53

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 1

ER -