Independence number and $[a,b]$-factors of graphs

Abstract

Let $G$ be a graph. The cardinality of any largest independent set of vertices in $G$ is called the independence number of $G$ and is denoted by $\alpha (G)$. Let $a$ and $b$ be integers with $0\le a\le b$. If $a=b$, it is assumed that $G$ be a connected graph, furthermore, $a\ge \alpha (G)$, $a/|V(G)|=0(\mathrm{mod}2)$ if $a$ is odd. We prove that every graph $G$ has an $[a,b]$-factor if its minimum degree is at least $(\frac{b+\alpha (G)a-\alpha (G)}{b})\lfloor \frac{b+\alpha (G)a}{2\alpha (G)}\rfloor -\frac{\alpha (G)}{b}(\lfloor \frac{b+\alpha (G)a}{2\alpha (G)}\rfloor {)}^2+\theta \frac{\alpha (G{)}^2}{b}+\frac{a}{b}\alpha (G)$, where $\theta =0$ if $a<b$, and $\theta =1$ if $a=b$. This degree condition is sharp.