Torus graph (${\mathbb{T}}_N=C_N\square C_N$)

For $N\ge 3$, the square torus graph has vertex set $(\mathbb{Z}/N\mathbb{Z}{)}^2$ and joins vertices differing by one in exactly one coordinate. It has $N^2$ vertices and $2N^2$ bonds. Translations act transitively on vertices; translations together with a quarter turn act transitively on bonds. A rectangle of side lengths at most $N-2$ has the same induced subgraph as its planar square lattice counterpart.