Contractions of 6-connected toroidal graphs

Let T6 denote the class of all 6-connected (equivalently 6-regular) toroidal graphs and let G∈T6 which is not minor-minimal in T6. Let G′∈T6 be a proper minor of G with maximum number of vertices. We show that |V(G)|−|V(G′)|=fw(G), where fw(G) denotes the face-width of the toroidal embedding of G. Consequently, we show that the only minor-minimal graphs in T6 are K7, K8−4K2, K9−C9, and K9−3K3.