A characterization of graphs having all \((g,f)\)-factors (Q1366610)

Let \(G\) be a graph with vertex set \(V\) and let \(g,f:V\to\mathbb{Z}^+\). We say that \(G\) has all \((g,f)\)-factors, if \(G\) has an \(h\)-factor for every \(h:V\to\mathbb{Z}^+\) such that \(g(v)\leq h(v)\leq f(v)\) for every \(v\in V\) and at least one such \(h\) exists. We derive from Tutte's \(f\)-factor theorem a similar characterization for the property of having all \((g,f)\)-factors. An analogous result for parity factors is presented also.