All fractional (g,f)-factors in graphs

Let $G$ be a graph, and $g,f:V(G)\rightarrow N$ be two functions with $g(x)\leq f(x)$ for each vertex $x$ in $G$. We say that $G$ has all fractional $(g,f)$-factors if $G$ includes a fractional $r$-factor for every $r:V(G)\rightarrow N$ such that $g(x)\leq r(x)\leq f(x)$ for each vertex $x$ in $G$. Let $H$ be a subgraph of $G$. We say that $G$ admits all fractional $(g,f)$-factors including $H$ if for every $r:V(G)\rightarrow N$ with $g(x)\leq r(x)\leq f(x)$ for each vertex $x$ in $G$, $G$ includes a fractional $r$-factor $F_h$ with $h(e)=1$ for any $e\in E(H)$, then we say that $G$ admits all fractional $(g,f)$-factors including $H$, where $h:E(G)\rightarrow [0,1]$ is the indicator function of $F_h$. In this paper, we obtain a characterization for the existence of all fractional $(g,f)$-factors including $H$ and pose a sufficient condition for a graph to have all fractional $(g,f)$-factors including $H$.

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