图有a,b因子的邻域条件

摘要

不含有子图K1,n的图称为K1,n-free图. 设G是一个具有顶点集V(G)的图. 设n(≥3), a和b是整数,使得b≥a≥1;若b是奇数,设b≥n-1. 我们证明了每个连通的K1,n-free图G在b|V(G)|为偶数,它的最小度至少是a+n-1, |V(G)|≥(2(a+b)-1)(a+b-1)/b,以及|NG(x)∪NG(y)|≥(a|V(G)|)/(a+b)对V的任意两个不邻接的点x和y都成立时,G有一个[a,b]因子.%A graph is called K1,n-free if it contains no K1,n as an induced subgraph. Let G be a graph with vertex set V(G). Let n(≥3), a and b be integers such that b≥a≥1, and if b is odd, bn-1. We prove that every K1,n-free connected graph G with b|V(G)|even has an [a,b]-factor if its minimum degree is at least a+n-1, |V(G)|≥(2(a+b)-1)(a+b-1)/b, and|NG(x)∪NG(y)|≥(a|V(G)|)/(a+b)for any two non-adjacent vertices x and y of V(G).