A Census of Edge-transitive Planar Tilings

Abstract

Recently, Graves, Pisanski, and Watkins have determined the growth rates of Bilinski diagrams of one-ended, 3-connected, edge-transitive planar maps. The computation depends solely on the edge-symbol $(p,q;k,l)$ that was introduced by B. Gr\”unbaum and G. C. Shephard in their classification of such planar tessellations. We present a census of such tessellations in which we describe some of their properties, such as whether the edge-transitive planar tessellation is vertex- or face-transitive, self-dual, bipartite, or Eulerian. In particular, we order such tessellations according to the growth rate and count the number of tessellations in each subclass.