Counting maps on doughnuts

How many maps with V vertices and E edges can be drawn on a doughnut with G holes? I solved this problem for doughnuts with up to 10 holes, and my colleagues Alain Giorgetti and Alexander Mednykh counted maps by number of edges alone on doughnuts with up to 11 holes. This expository paper outlines, in terms meant to be understandable by a non-specialist, the methods we used and those used by other researchers to obtain the results upon which our own research depends.