Enumeration of r-regular maps on the torus. Part I: Rooted maps on the torus,the projective plane and the Klein bottle. Sensed maps on the torus

The work that consists of two parts is devoted to the problem of enumerating unrooted $r$-regular maps on the torus up to all its symmetries. We begin with enumerating near-$r$-regular rooted maps on the torus, the projective plane and the Klein bottle, as well as some special kinds of maps on the sphere: near-$r$-regular maps, maps with multiple leaves and maps with multiple root darts. For $r=3$ and $r=4$ we obtain exact analytical formulas. For larger $r$ we derive recurrence relations. Then we enumerate $r$-regular maps on the torus up to homeomorphisms that preserve its orientation — so-called sensed maps. Using the concept of a quotient map on an orbifold we reduce this problem to enumeration of certain above-mentioned classes of rooted maps. For $r=3$ and $r=4$ we obtain closed-form expressions for the numbers of $r$-regular sensed maps by edges. All these results will be used in the second part of the work to enumerate $r$-regular maps on the torus up to all homeomorphisms — so-called unsensed maps.