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

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 $k=2$.