Title: Semi-equivelar maps on the torus are Archimedean,
Abstract: If the face-cycles at all the vertices in a map on a surface are of same type
then the map is called semi-equivelar. There are eleven types of Archimedean
tilings on the plane. All the Archimedean tilings are semi-equivelar maps. If a
map $X$ on the torus is a quotient of an Archimedean tiling on the plane then
the map $X$ is semi-equivelar. We show that each semi-equivelar map on the
torus is a quotient of an Archimedean tiling on the plane.
Vertex-transitive maps are semi-equivelar maps. We know that four types of
semi-equivelar maps on the torus are always vertex-transitive and there are
examples of other seven types of semi-equivelar maps which are not
vertex-transitive. We show that the number of ${\rm Aut}(Y)$-orbits of vertices
for any semi-equivelar map $Y$ on the torus is at most six. In fact, the number
of orbits is at most three except one type of semi-equivelar maps. Our bounds
on the number of orbits are sharp.