A class of maps on the torus and their vertex orbits

Abstract

A tiling (edge-to-edge) of the plane is a family of tiles that cover the plane without gaps or overlaps. Vertex figure of a vertex in a tiling to be the union of all edges incident to that vertex. A tiling is $k$-vertex-homogeneous if any two vertices with congruent vertex figures are symmetric with each other and the vertices form precisely $k$ transitivity classes with respect to the group of all symmetries of the tiling. In this article, we discuss that if a map is the quotient of a plane's $k$-vertex-homogeneous lattice ($k\ge 4$) then what would be the sharp bounds of the number of vertex orbits.