A neighborhood condition for graphs to have restricted fractional (g,f)-factors

Abstract

Let $h$ be a function defined on $E(G)$ with $h(e)\in[0,1]$ for any $e\in E(G)$. Set $d_G^{h}(x)=\sum_{e\ni x}h(e)$. If $g(x)\leq d_G^{h}(x)\leq f(x)$ for every $x\in V(G)$, then we call the graph $F_h$ with vertex set $V(G)$ and edge set $E_h$ a fractional $(g,f)$-factor of $G$ with indicator function $h$, where $E_h=\{e:e\in E(G),h(e)>0\}$. Let $M$ and $N$ be two sets of independent edges of $G$ with $M\cap N=\emptyset$, $|M|=m$ and $|N|=n$. If $G$ admits a fractional $(g,f)$-factor $F_h$ such that $h(e)=1$ for any $e\in M$ and $h(e)=0$ for any $e\in N$, then we say that $G$ has a fractional $(g,f)$-factor with the property $E(m,n)$. In this paper, we present a neighborhood condition for the existence of a fractional $(g,f)$-factor with the property $E(1,n)$ in a graph. Furthermore, it is shown that the neighborhood condition is sharp.