It is demonstrated that the problems of balancing a reinsurance network and finding the maximum flow in a graph are identical. Gale's theorem is applied first in order to prove a conjecture of Sousselier concerning simple first order networks, next to extend those results to any network. The balanced reinsurance scheme can effectively be constructed by means of Ford and Fulkerson's algorithm, as is shown by an example. Finally, we demonstrate that graph theory provides tools able to tackle more general problems than the one considered by Sousselier: capacity constraints and reinsurance costs are introduced in the model.