@book{d01108e0d5854a07b7284acd678bcd5a,

title = "A fan 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. $\max\{d^w(x),d^w(y)\mid d(x,y)=2\}\geq c/2$; 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 $c$. This generalizes a theorem of Fan on the existence of long cycles in unweighted graphs to weighted graphs. We also show we cannot omit Condition 2 or 3 in the above result.",

keywords = "MSC-05C45, MSC-05C35, MSC-05C38",

author = "Hajo Broersma and Shenggui Zhang and Xueliang Li and Ligong Wang",

year = "2000",

language = "English",

series = "Memorandum Faculteit TW",

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

number = "1513",

}