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Article

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

Discrete Mathematics. 2019. Vol. 342. No. 2. P. 584-599.
Omelchenko A., Краско Е. С.

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.