Enumeration of loopless maps on the projective plane

In this paper we study the rooted loopless maps on the sphere and the projective plane with the valency of root-face and the number of edges as parameters. Explicit expression of enumerating function is obtained for such maps on the sphere and the projective plane. A parametric expression of the generating function is obtained for such maps on the projective plane, from which asymptotic evaluations are derived.