The Crossing Number of Two-Maps on Orientable Surfaces

In this paper,we discuss the crossing numbers of two one-vertex maps on orientable surfaces.By using a reductive method,we give the crossing number of two one-vertex maps with one face on an orientable surface and the crossing number of a one-vertex map with one face and a one-vertex map with two faces on an orientable surface.This provides a lower bound for the crossing number of two general maps on an orientable surface.