Stability Number and Fractional $F$-Factors in Graphs

Abstract

Let $G$ be a graph with vertex set $V(G)$ and let $f$ be a nonnegative integer-valued function defined on $V(G)$. A spanning subgraph $F$ of $G$ is called a fractional $f$-factor if $d_G^h(x)=f(x)$ for every $x\in V(F)$. In this paper, we prove that if $\delta (G)\ge b$ and $\alpha (G)\le \frac{4a(\delta -b)}{(b+1{)}^2}$, then $G$ has a fractional $f$-factor. Where $a$ and $b$ are integers such that $0\le a\le f(x)\le b$ for every $x\in V(G)$. Therefore, we prove that the fractional analogue of Conjecture in $[2]$ is true.