Complete factors and \(f\)-factors (Q1567623)

Let \(G\) be a graph and \(F\) a spanning subgraph of \(G\) with at least two components and with all components complete. Let \(f\) be an integer-valued function defined on \(V(G)\) with \(\sum_{x \in V(G)} f(x)\) even. If for each component \(C\) of \(F\), \(G - V(C)\) has an \(f\)-factor (that is, a spanning subgraph \(H\) in which \(\deg_H(x) = f(x)\) for all \(x \in V(G)\)), then \(G\) has an \(f\)-factor.