A Toughness Condition for the Existence of $f$-Factors in Graphs

Abstract

Let $G$ be a graph and let $f$ be a positive integer-valued function defined on $V(G)$ such that $1\le a\le f(x)\le b\le 2a$ for every $x\in V(G)$. If $t(G)\ge \frac{b^2}{a}$, $|V(G)|\ge \frac{b^2}{a}+1$, and $f(V(G))$ is even, then $G$ has an $f$-factor.