Enumerating near-4-regular maps on the sphere and the torus

In this paper rooted near-4-regular maps on the plane and the torus are cou nted with formulae with respect to four parameters: the root valency, the n umber of edges, the inner faces, and nonroot-vertex loops. In particular, t he number of noted near-4-regular maps on those surfaces with exactly k non root-vertex loops is investigated. (C) 2001 Elsevier Science B.V. All right s reserved.