Quasi-vertex-transitive maps on the plane

Abstract

Quasi-vertex-transitive maps are the homogeneous maps on the plane with finitely many vertex orbits under the action of their automorphism groups. We show that there exist quasi-vertex-transitive maps of the types [p3,3] for p≡1 (mod 6), but there does not exist vertex-transitive map of such types. In particular, we determine the surface with the lowest possible genus that admit a polyhedral map of type [53,3].