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 4) then what would be the sharp bounds of the number of vertex orbits.
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