Heavy Cycles in 2-connected Triangle-free Weighted Graphs  

作　　者：Xue Zheng LV Pei WANG 

机构地区：[1]Department of Mathematics, Renmin University of China [2]Department of Mathematics and Physics, China University of Petroleum

出　　处：《Acta Mathematica Sinica,English Series》2015年第10期1555-1562,共8页数学学报（英文版）

基　　金：Supported by National Natural Science Foundation of China(Grant No.11001269)

摘　　要：A weighted graph is one in which every edge e is assigned a nonnegative number, called the weight of e. The sum of the weights of the edges incident with a vertex v is called the weighted degree of v, denoted by dw（v）. The weight of a cycle is defined as the sum of the weights of its edges. Fujisawa proved that if G is a 2-connected triangle-free weighted graph such that the minimum weighted degree of G is at least d, then G contains a cycle of weight at least 2d. In this paper, we proved that if G is a 2-connected triangle-free weighted graph of even size such that dw（u） ＋ dw（v） ≥ 2d holds for any pair of nonadjacent vertices u, v ∈ V（G）, then G contains a cycle of weight at least 2d.A weighted graph is one in which every edge e is assigned a nonnegative number, called the weight of e. The sum of the weights of the edges incident with a vertex v is called the weighted degree of v, denoted by dw（v）. The weight of a cycle is defined as the sum of the weights of its edges. Fujisawa proved that if G is a 2-connected triangle-free weighted graph such that the minimum weighted degree of G is at least d, then G contains a cycle of weight at least 2d. In this paper, we proved that if G is a 2-connected triangle-free weighted graph of even size such that dw（u） ＋ dw（v） ≥ 2d holds for any pair of nonadjacent vertices u, v ∈ V（G）, then G contains a cycle of weight at least 2d.

关 键 词：Heavy cycles triangle-free graphs weighted graphs 

分 类 号：O157.5[理学—数学] TP311.13[理学—基础数学]