Triangulations on closed surfaces which quadrangulate other surfaces II

It has already been proved that given two closed surfaces F12 and F22 with 2χ(F12)-χ(F22)⩾4, there exists a triangulation on F12 which can be embedded on F22 as a quadrangulation. In this paper we refine that result, showing that there exists an integer g0 such that for any two closed surfaces with genus g1⩾g0 and genus g2 satisfying 2χ(F12)-χ(F22)⩾O(g1), there exists a triangulation of the first surface which can be re-embedded on the second as a quadrangulation. Moreover, on the right-hand side of the inequality, we obtain a concrete expression which is asymptotically O(g1). We also obtain similar results for non-orientable surfaces.