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

Abstract

Let $a$ and $b$ be integers such that $1\le a<b$, and let $G$ be a graph of order $n$ with $n\ge \frac{(a+b)(2a+2b-3)}{a+1}$ and the minimum degree $\delta (G)\ge \frac{(b-1{)}^2-(a+1)(b-a-2)}{a+1}$. Let $g(x)$ and $f(x)$ be two nonnegative integer-valued functions defined on $V(G)$ such that $a\le g(x)\le f(x)\le b$ for each $x\in V(G)$. We prove that if $|N_G(x)\cup N_G(y)|\ge \frac{(b-1)n}{a+b}$ for any two nonadjacent vertices $x$ and $y$ in $G$, then $G$ has a $(g,f)$-factor. Furthermore, it is shown that the result in this paper is best possible in some sense.