In this paper the number of rooted (near-) 4-regular maps on the projective plane are investigated with respect to the root-valency, the number of edges, the number of inner faces, the number of nonroot-vertex-loops, the number of nonroot-vertexblocks. As special cases, formulae for several types of rooted 4-regular maps such as 2-connected 4-regularâ€¦ (More)