  We perform the asymptotic enumeration of two classes of rooted maps on
orientable surfaces of genus g: m-hypermaps and m-constellations. For m=2, they
correspond respectively to maps with even face degrees and bipartite maps. We
obtain explicit asymptotic formulas for the number of such maps with any finite
set of allowed face degrees.
  Our proofs rely on the generalisation to orientable surfaces of the
Bouttier-Di Francesco-Guitter bijection, and on generating series methods. We
show that each of the 2g fondamental cycles of the surface contributes a factor
m between the numbers of m-hypermaps and m-constellations -- for example, large
maps of genus g with even face degrees are bipartite with probability tending
to 1/2^{2g}.
  A special case of our results implies former conjectures of Gao.
