Orientation with degree bounds

Theorem (Frank and Gyárfás ). Let G=(V,E) be an undirected graph, and let [math]l_v \leq u_v[/math] be nonnegative integers for every [math]v \in V[/math]. G has an orientation D satisfying [math]l_v \leq \varrho_D(v) \leq u_v[/math] for every [math]v \in V[/math] if and only if every node set [math]X \subseteq V[/math] spans at most u(X) edges and is incident with at least l(X) edges of G.

The version with only lower bounds was proved by Hakimi . The theorem is a consequence of Hoffman's circulation theorem, but it can also be proved easily directly.