Minimal covers of equivelar toroidal maps - Drach, Kostiantyn (Ars mathematica contemporanea, 2015)
Given any equivelar map on the torus, it is natural to consider its covering maps. The most basic of these coverings are finite toroidal maps or infinite tessellations of the Euclidean plane. In this paper, we prove that each equivelar map on the torus has a unique minimal toroidal rotary cover and also a unique minimal toroidal regular cover. That is to say, of all the toroidal rotary (or regular) maps covering a given map, there is a unique smallest. Furthermore, using the Gaussian and Eisenstein integers, we construct these covers explicitly.
Keywords: Eisenstein integers; Eisensteinova cela števila; Gaussian integers; Gaussova cela števila; minimal covers; minimalni krovi; regular maps; regularni zemljevidi; rotarijski zemljevidi; rotary maps