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 \( 1 \leq a \leq f(x) \leq b \leq 2a \) for every \( x \in V(G) \). If \( t(G) \geq \frac{b^2}{a} \), \( |V(G)| \geq \frac{b^2}{a} + 1 \), and \( f(V(G)) \) is even, then \( G \) has an \( f \)-factor.