Growth of Edge-Homogeneous Tessellations

A tessellation is understood to be a 1-ended, locally finite, 3-connected planar map. The edge-symbol $\langle p,q;k,\ell\rangle$ of an edge of a tessellation $T$ is a 4-tuple listing the valences $p$ and $q$ of its two incident vertices and the covalences $k$ and $\ell$ of its two incident faces. To say that $T$ is edge-homogeneous means that all edges of $T$ have the same edge-symbol. By a result of Grunbaum and Shephard, each edge-transitive tessellation may be identified with its edge-symbol. It is shown that the growth rate of $T$ is given by a function $g(t)=\frac12(t-2+\sqrt{t^2-4t})$ of the single variable $t=(\frac{p+q}2-2)(\frac{k+\ell}2-2)$, except that the growth rate equals $g(t-1)$ when the edge-symbol of $T$ or its planar dual has the form $\langle3,q;4,4\rangle$, where $q\geq6$. Thus, for each integer $t\geq4$, there are only finitely many edge-homogeneous tessellations whose growth rate equals $g(t)$, allowing a complete list of such tessellations to be compiled in terms of increasing growth rate. The maximum value of the quantity $\frac1p+\frac1q+\frac1k+\frac1\ell$ for tessellations with given value $t$ is shown to decrease monotonically as $t$ increases, while the minimum value decreases only asymptotically. Methods are demonstrated for concrete enumeration of the sets of faces and vertices at any given facial distance from a fixed face, edge, or vertex.