Graphs that triangulate a given surface and quadrangulate another surface

We show that for any closed surface F with χ ( F ) ⩽ − 4 (or χ ( F ) ⩽ − 2 ), there exist graphs that triangulate the torus or the Klein bottle (or the projective plane) and that quadrangulate F. We also give a sufficient condition for a graph triangulating a closed surface to quadrangulate some other surface.