关于赋权图中重圈的一个范型定理

A Fan Type Theorem for Heavy Cycles in Weighted Graphs

设G=(V,E;w)为赋权图,定义G中点v的权度d_G^w(v)为G中与v相关联的所有边的权和.该文证明了下述定理:假设G为满足下列条件的2-连通赋权图:(i)对G中任何导出路xyz都有w(xy)=w(yz);(ii)对G中每一个与K_(1,3)或K_(1,3+e)同构的导出子图T,T中所有边的权都相等并且min{max{d_G^w(x),D_G^w(y)}:d(x,y)=2,x,y∈V(T)}≥c/2.那么,G中存在哈密尔顿圈或者存在权和至少为c的圈.该结论分别推广了Fan,Bedrossian等人和Zhang等人的相关定理.

Let G = （V, E; w） be a weighted graph, and define the weighted degree dw/G（v） of a vertex v in G as the sum of the weights of the edges incident with v. In this paper, the following theorem is proved： suppose G is a 2-connected weighted graph, where （i） w（xy） = w（yz） for every induced path xyz, and （ii） in every induced subgraph T of G isomorphic to K1,3 or K1,3 ＋ e, all the edges of T have the same weight and min{max{dw/G（x）,dw/G（y）} ： d（x,y） = 2, x, y ∈ V（T）} ≥ c/2, then G contains either a Hamilton cycle or a cycle of weight c at least. This respectively generalizes three theorems of Fan, Bedrossian et al and Zhang et al.