We can prove Hall’s marriage theorem as well as Konig-Egervary theorem using max-flow min-cut or Menger’s theorem. See exercises in the next section. Also, see Section 6.10.
The max-flow min-cut theorem can be derived from a more powerful theorem called the strong duality theorem in linear programming (see https://en.wikipedia.org/wiki/Max-flow_min-cut_theorem#Linear_program_formulation). This latter theorem for example can be used to prove the Monge-Kantorovich duality theorem in Optimal transport theory (https://en.wikipedia.org/wiki/Transportation_theory_(mathematics)).