On min-max theorems in matching theory

Frank, Sebö and Tardos [2] proved that for any connected bipartite graft (G,T), the minimum size of a T-join is equal to the maximum value of a partition of A, where A is one of the two colour classes of G. Their proof consists of constructing a partition of A of value ∣F∣, by using a minimum T-join F. That proof depends heavily on the properties of distances in graphs with conservative weightings. We follow the dual approach, that is starting from a partition of A of maximum value k, we construct a T-join of size k. Our proof relies only on Tutteʹs theorem [9] on perfect matchings. known [3] that the results of Edmonds-Johnson [1], Lovász [5] on 2-packing of T-cuts, of Seymour [7,8] on packing of T-cuts in bipartite graphs and in grafts that cannot be T-contracted onto (K4, V(K4)), and of Sebö [6] on packing of T-borders are implied by this theorem of Frank et al. in contribution of the present paper is that all of these results can be derived from Tutteʹs theorem.