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

Haitze J. Broersma, S. Zhang, X. Li, Xueliang Li

Research output: Book/ReportReportProfessional

22 Downloads (Pure)


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 $k=2$.
Original languageUndefined
Place of PublicationEnschede
PublisherUniversity of Twente, Department of Applied Mathematics
Number of pages12
Publication statusPublished - 2000

Publication series

NameMemorandum Faculteit TW
PublisherUniversity of Twente
ISSN (Print)0169-2690


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

Cite this