Rooted General Maps on All Surfaces

Abstract

In this paper, we concentrate on rooted general maps on all surfaces(orientable and nonorientable) without regard to genus and present the enumerating equation with respect to vertices and edges, which is a Riccati’s equation. To solve it, a new solution in continued fraction form is given. As two especial cases, the corresponding results of rooted general maps and rooted monopole maps on all surfaces with respect to edges regardless of genus are obtained.