The torus grid graph T_(m, n) is the graph formed from the graph Cartesian product C_m square C_n of the cycle graphs C_m and C_n. C_m square C_n is isomorphic to C_n square C_m. C_m square C_n can be formed starting with an m×n grid graph and connecting corresponding left/right and top/bottom vertex pairs with edges. While such an embedding has overlapping edges in the plane, it can naturally be placed on the surface of a torus with no edge intersections or overlaps. Torus grid graphs are therefore toroidal graphs. The isomorphic torus grid graphs C_10 square C_6 and C_6 square C_10 are illustrated above. The torus grid graphs are quartic and Hamiltonian and have vertex count left bracketing bar C_m square C_n right bracketing bar = m n.

graph Cartesian product | grid graph | toroidal graph