The Quadrangulation Conjecture for Orientable Surfaces

By means of character theory and symmetric functions, D. M. Jackson and T. I. Visentin (1990, Trans. Amer. Math. Soc.322, (343–363)) proved the existence of certain bijections between the set of quadrangulations in orientable surfaces and decorated maps (with marked edges and coloured vertices) in orientable surfaces. The bijections preserve a weight function consisting of a pair (g,n) of integer parameters. For quadrangulations, g is the genus and n is the number of faces. For decorated maps, g is the genus plus half the number of white vertices and n is the number of edges. The Quadrangulation Conjecture concerns the problem of finding a natural bijection of this type. Tutteʹs medial construction is a solution in the special case g=0 of planar maps. We give a construction of a bijection Ξ̃ which both extends Tutteʹs medial construction to non-planar maps and preserves the parameter n of the Quadrangulation Conjecture. (The parameter g is not generally preserved, except when g=0.) Non-orientable surfaces play an important part in the construction of Ξ̃. As part of the construction, we introduce a bijection between orientable rooted quadrangulations and locally orientable, bipartite, rooted quadrangulations.