Enumerating near-4-regular maps on the sphere and the torus Original Research Article

In this paper rooted near-4-regular maps on the plane and the torus are counted with formulae with respect to four parameters: the root valency, the number of edges, the inner faces, and nonroot-vertex loops. In particular, the number of rooted near-4-regular maps on those surfaces with exactly k nonroot-vertex loops is investigated.