The problem of matrices with given row and column sums is well known to be a network flow problem. First the matrix problem. Perhaps the nicest generalization of the fundamental Gale-Ryser Theorem is the following result of my Ph. But they also have helped in other ways.
Many problems here relate to degree constrained subgraph problems.
R.P. Anstee, Invariant Sets of Arcs in Network Flow Problems, Discrete Appl.