Structural Remarks on Bipartite Graphs with Unique f-Factors

In this note we will derive some structural results for a bipartite graph G with a unique f-factor. Two necessary conditions will be that G is saturated, meaning that the addition of any edge leads to a second f-factor, and that f_A, f_B≥1. Here f_A and f_B are defined as the minimum of f over the vertices in the two partite sets A and B of G, respectively. Our main result states that G has at least f_A + f_B vertices for which d_G (v) = f(v) holds.