关于图的分数k-因子存在性的一些结果

设g(x)≤f(x)是定义在V(G)上的两个整数值函数,h(e)∈0,1]是定义在图G的边集E(G)上的函数。令dGh(x)=移e∈Exh(e),其中Ex={xy:xy∈E(G)}。若对所有的x∈V(G)都有g(x)≤dGh(x)≤f(x)成立,称h是G的一个(g,f)-表示函数。Gh是图G的一个支撑子图使得E(Gh)={e:e∈E(G),h(e)≠0},则称Gh是G的一个分数(g,f)-因子。文章给出,若对V(G)中的任意两个顶点u和v,G-{u,v}有分数k-因子存在。则G有一个分数k-因子不含图G中任意给定的边e∈E(G);当G有分数1-因子F=Gh存在时,对任意e∈F,G-V(e)有分数k-因子存在,则G有分数k-因子。

Some Results on Fractional K-factors of Graphs

Let g(x)≤f(x) be two integral-valued functions defined on V(G) and h(e)∈ be a function defined on E(G).Let d(x)=∑ h(e) where E ={xy:xy E(G)}.We call h an indicator function if g(x)(x) f(x) holds for each x V(G).Let G be a spanning sub-graph of G such that E(G)={e:e E(G),h(e) 0},We call G a fractional(g,f)-factor.The main results in this paper are:if for any two vertices u and v of V(G),G-{u,v} has fractional k-factor.Then G has a fractional k-factor excluding any given edge e.If G has a fractional 1-factor F=G,G-V(e) has fractional k-factors for any e F,then G has a fractional k-factor.