Hoffman's circulation theorem
Let [math]D=(V,A)[/math] be a digraph. a circulation in [math]D[/math] is a function [math]x: A \to {\mathbb R}[/math] for which [math]\varrho_x(v)=\delta_x(v)[/math] for every [math]v\in V[/math] (see Help:Notation). The following theorem was proved by Hoffman in 1956 and cited in [1].
Theorem (Hoffman [1]). [math]Let D=(V,A)[/math] be a digraph, with lower bound [math]l: A \to {\mathbb R}[/math] and upper bounds [math]u: A \to {\mathbb R}[/math] on the edges such that [math]l \leq u[/math]. There exists a circulation [math]x[/math] satisfying [math]l(a) \leq x(a) \leq u(a)[/math] for every [math]a \in A[/math] if and only if
[math]\varrho_f(Z) \leq \delta_g(Z) \text{ for every } Z \subseteq V.[/math]
If [math]u[/math] and [math]l[/math] are integer-valued, then [math]x[/math] can be integer-valued too.
