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 \(\delta(G) \geq b\) and \(\alpha(G) \leq \frac{4a(\delta-b)}{(b+1)^2}\), then \(G\) has a fractional \(f\)-factor. Where \(a\) and \(b\) are integers such that \(0 \leq a \leq f(x) \leq b\) for every \(x \in V(G)\). Therefore, we prove that the fractional analogue of Conjecture in \([2]\) is true.