Counting rooted nearly cubic planar maps

Abstract

In this paper it is provided that a functional equation for enumerating rooted nearly cubic planar maps with the valency of root-vertex, the size and the valency of root-face of the maps as three parameters. In particular, the number of rooted nearly cubic planar maps with the valency of root-vertex and the size as two parameters, as an answer to open problem in 1989, can be also derived.