@book{37bb88512d8c43d4941ada9f3974c9db,

title = "A $\sigma_3$ type condition for heavy cycles in weighted graphs",

abstract = "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 $k=2$.",

keywords = "MSC-05C35, MSC-05C38, EWI-3334, METIS-141192, IR-65702, MSC-05C45",

author = "Broersma, {Haitze J.} and S. Zhang and X. Li and Xueliang Li",

note = "Imported from MEMORANDA",

year = "2000",

language = "Undefined",

series = "Memorandum Faculteit TW",

publisher = "University of Twente, Department of Applied Mathematics",

number = "1514",

}