  A 2-uniform tiling is an edge-to-edge tiling by regular polygons having $2$
distinct transitivity classes of vertices. There are 20 distinct 2-uniform
tilings (these are of $14$ different types) on the plane, and since the plane
is the universal cover of the torus, it is natural to explore maps on the torus
that correspond to the 2-uniform tilings. In this article, we discuss that if a
map is the quotient of a plane's $2$-uniform lattice then what would be the
bounds of the number of vertex orbits.
  A 3-uniform tiling is an edge-to-edge tiling by regular polygons having $3$
distinct transitivity classes of vertices. There are $61$ distinct $3$-uniform
tilings on the plane. In this article, we discuss that if a map is the quotient
of a plane's $3$-uniform lattice then what would be the bounds of the number of
vertex orbits.
