Enumeration of rooted 4-regular maps without planar loops

With the help of composition formula we obtain a very simple expression that enables us to find the generating function for rooted 4-regular maps without planar loops on an arbitrary orientable or non-orientable surface of a given genus g, assuming that the generating function for all rooted 4-regular maps on this surface is known. As an application of this result, we derive explicit expressions for the corresponding generating functions for maps on the sphere, the projective plane, the torus, and the Klein bottle. We also provide an explicit formula for the number of unlabelled 4-regular maps without planar loops on the sphere, counted up to orientation-preserving homeomorphisms.