图的独立数与分数一致性

Independent numbers of graphs and fractional uniform graphs

设G是一个顶点集为V(G),最小度为δ(G),独立数为α(G)的图,k≥2是整数。图G的支撑子图F称作是图G的分数k-因子,如果对于每一个x∈V(F)都有dh G(x)=k。如果对于图G的每条边e,图G都有一个分数k-因子包含它而且同时有一个分数k-因子不包含它,则称图G为分数k一致图。证明了如果δ(G)≥k+2,且α(G)≤4k(δ-k-1)/(k+1)2,则图G是一个分数k一致图。

Let G be a graph with vertex set V（ G） , minimum degreeδ（ G） and independent numberα（ G） .Let k≥2 be an integer.A spanning subgraph F of G is called a fractional k-factor if dhG（x） =k for every x∈V（F）.A graph G is called a fractional k-uniform graph if for each edge of G, there is a fractional k-factor containing it and another one ex-cluding it.In this paper, we prove that if δ（G）≥k＋2 and α（G）≤4k（δ-k-1）（k＋1）2 , then G is a fractional k-uniform graph.