Max-Flow Min-Cut Theorem

If $f$ is a flow in a flow network $G=(V,E)$ with source $s$ and sink $t$, then the following conditions are equivalent:

1. $f$ is a maximum flow in $G$.
2. The residual network $G_f$ contains no augmenting paths.
3. $|f|=c(S,T)$ for some cut $(S,T)$ of $G$.

Proof

See CLRS p. 723.