图中存在f-因子的度条件

设n≥3是一个整数，G是一个具有顶点集V(G)的图．并设，是定义在V(G)上的非负整值函数．设a=mx|g(x)|x∈V(G)|，b=min|f(x)|x∈V(G)|，并有b，a≥2，n≥b／(a-1) 1，如果存在点v∈V(G)使得f(v)m|(mod 2)，假定b≥n-1．则每个连通的使得f(V(G))为偶数的K1,a-free图G有f-因子，如果它的最小度至少是((n-1)(b 1) a)/a)b(n-1) a/2(n-1)] (n-1)/a](b(n-1) a/2(n-1)])^2 n-3.

A degree condition for graphs to have f-factors

Let n≥3 be an integer and G be a graph. Let f(x) be an integer-valued function defined on V(G), and let a=min{f(x):x∈V(G)},b=max{f(x):x∈V(G)}, and b, a≥2, n≥a/(b-1)+1,if there is a vertex x∈V(G) such that f(x) odd, we assume that a≥n-1. We prove that every connected graph G with f(V(G)) even has an f-factor if it contains no K1,n as an induced subgraph and its minimum degree is at least((n-1)(b+1)+a)/(a)(b(n-1)+a)/(2(n-1))-(n-1)/(a)(b(n-1)+a)/(2(n-1))2+n-3.