### Abstract

Original language | Undefined |
---|---|

Place of Publication | Enschede |

Publisher | University of Twente, Department of Applied Mathematics |

Number of pages | 12 |

Publication status | Published - 2000 |

### Publication series

Name | Memorandum Faculteit TW |
---|---|

Publisher | University of Twente |

No. | 1514 |

ISSN (Print) | 0169-2690 |

### Keywords

- MSC-05C35
- MSC-05C38
- EWI-3334
- METIS-141192
- IR-65702
- MSC-05C45

### Cite this

*A $\sigma_3$ type condition for heavy cycles in weighted graphs*. (Memorandum Faculteit TW; No. 1514). Enschede: University of Twente, Department of Applied Mathematics.

}

*A $\sigma_3$ type condition for heavy cycles in weighted graphs*. Memorandum Faculteit TW, no. 1514, University of Twente, Department of Applied Mathematics, Enschede.

**A $\sigma_3$ type condition for heavy cycles in weighted graphs.** / Broersma, Haitze J.; Zhang, S.; Li, X.; Li, Xueliang.

Research output: Book/Report › Report › Professional

TY - BOOK

T1 - A $\sigma_3$ type condition for heavy cycles in weighted graphs

AU - Broersma, Haitze J.

AU - Zhang, S.

AU - Li, X.

AU - Li, Xueliang

N1 - Imported from MEMORANDA

PY - 2000

Y1 - 2000

N2 - A weighted graph is a graph in which each edge $e$ is assigned a non-negative number $w(e)$, called the weight of $e$. The weight of a cycle is the sum of the weights of its edges. The weighted degree $d^w(v)$ of a vertex $v$ is the sum of the weights of the edges incident with $v$. In this paper, we prove the following result: Suppose $G$ is a 2-connected weighted graph which satisfies the following conditions: 1. The weighted degree sum of any three independent vertices is at least $m$; 2. $w(xz)=w(yz)$ for every vertex $z\in N(x)\cap N(y)$ with $d(x,y)=2$; 3. In every triangle $T$ of $G$, either all edges of $T$ have different weights or all edges of $T$ have the same weight. Then $G$ contains either a Hamilton cycle or a cycle of weight at least $2m/3$. This generalizes a theorem of Fournier and Fraisse on the existence of long cycles in $k$-connected unweighted graphs in the case $k=2$. Our proof of the above result also suggests a new proof to the theorem of Fournier and Fraisse in the case $k=2$.

AB - A weighted graph is a graph in which each edge $e$ is assigned a non-negative number $w(e)$, called the weight of $e$. The weight of a cycle is the sum of the weights of its edges. The weighted degree $d^w(v)$ of a vertex $v$ is the sum of the weights of the edges incident with $v$. In this paper, we prove the following result: Suppose $G$ is a 2-connected weighted graph which satisfies the following conditions: 1. The weighted degree sum of any three independent vertices is at least $m$; 2. $w(xz)=w(yz)$ for every vertex $z\in N(x)\cap N(y)$ with $d(x,y)=2$; 3. In every triangle $T$ of $G$, either all edges of $T$ have different weights or all edges of $T$ have the same weight. Then $G$ contains either a Hamilton cycle or a cycle of weight at least $2m/3$. This generalizes a theorem of Fournier and Fraisse on the existence of long cycles in $k$-connected unweighted graphs in the case $k=2$. Our proof of the above result also suggests a new proof to the theorem of Fournier and Fraisse in the case $k=2$.

KW - MSC-05C35

KW - MSC-05C38

KW - EWI-3334

KW - METIS-141192

KW - IR-65702

KW - MSC-05C45

M3 - Report

T3 - Memorandum Faculteit TW

BT - A $\sigma_3$ type condition for heavy cycles in weighted graphs

PB - University of Twente, Department of Applied Mathematics

CY - Enschede

ER -