Periodic square cellulation of the torus

Fix an integer $L\ge 3$ and write ${\mathbb{Z}}_L=\mathbb{Z}\mathrm{/}L\mathbb{Z}$. The periodic square cellulation has vertices $q_{i,j}$, horizontal edges $h_{i,j}$ from $q_{i,j}$ to $q_{i+1,j}$, vertical edges $v_{i,j}$ from $q_{i,j}$ to $q_{i,j+1}$, and faces $f_{i,j}$, for $i,j\in {\mathbb{Z}}_L$. Every index is read modulo $L$. The boundary of $f_{i,j}$ consists of $h_{i,j},v_{i+1,j},h_{i,j+1},v_{i,j}$. Equivalently, this is the quotient of the unit square tiling of ${\mathbb{R}}^2$ by translations in $L{\mathbb{Z}}^2$.

The cellulation has $L^2$ vertices, $2L^2$ edges, and $L^2$ faces. The condition $L\ge 3$ ensures that the four edges in each star and face are distinct, and that every edge has two distinct endpoints and two distinct incident faces. The vertex-edge graph joins vertices joined by an edge. The face-adjacency graph joins faces sharing an edge; both graphs are connected.