A Degree Condition for Graphs to Have $(g,f)$-Factors

Abstract

Let $G$ be a graph of order $n$, and let $a$ and $b$ be integers such that $1\le a<b$. Let $g(x)$ and $f(x)$ be two nonnegative integer-valued functions defined on $V(G)$ such that $a\le g(x)<f(x)\le b$ for each $x\in V(G)$. Then $G$ has a $(g,f)$-factor if the minimum degree $\delta (G)\ge \frac{(b-1{)}^2-(a+1)(a+b-1)}{a+1}$ ,$n>\frac{(a+b)(a+b-1)}{a+1}$ and $max\{d_G(x),d_G(y)\}\ge \frac{(b-1)n}{a+b}$ for any two nonadjacent vertices $x$ and $y$ in $G$. Furthermore, it is shown that the result in this paper is best possible in some sense.