Bounding Tree-Width via Contraction on the Projective Plane and Torus

If X is a collection of edges in a graph G, let G/X denote the contraction of X. Following a question of Oxley and a conjecture of Oporowski, we prove that every projective-planar graph G admits an edge-partition {X,Y } such that G/X and G/Y have tree-width at most three. We prove that every toroidal graph G admits an edge-partition {X,Y } such that G/X and G/Y have tree-width at most three and four, respectively.