Contractions of 6-connected toroidal graphs

Let T 6 denote the class of all 6-connected (equivalently 6-regular) toroidal graphs and let G ∈ T 6 which is not minor-minimal in T 6 . Let G ′ ∈ T 6 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 T 6 are K 7 , K 8 − 4 K 2 , K 9 − C 9 , and K 9 − 3 K 3 .