C_4-face-magic toroidal labelings on C_m × C_n

For a graph G = (V, E) naturally embedded in the torus, let ℱ(G) denote the set of faces of G. Then, G is called a Cn-face-magic toroidal graph if there exists a bijection f : V(G) → {1, 2, …, |V(G)|} such that for every F ∈ ℱ(G) with F ≅ Cn, the sum of all the vertex labels along Cn is a constant S. Let xv = f(v) for all v ∈ V(G). We call {xv : v ∈ V(G)} a Cn-face-magic toroidal labeling on G. We show that, for all m, n ≥ 2, Cm × Cn admits a C4-face-magic toroidal labeling if and only if either m = 2, or n = 2, or both m and n are even. We say that a C4-face-magic toroidal labeling {xi, j : (i, j) ∈ V(C2m × C2n)} on C2m × C2n is antipodal balanced if $x_{i,j} + x_{i+m,j+n} = \tfrac{1}{2} S$, for all (i, j) ∈ V(C2m × C2n). We show that there exists an antipodal balanced C4-face-magic toroidal labeling on C2m × C2n if and only if the parity of m and n are the same. Furthermore, when both m and n are even, an antipodal balanced C4-face-magic toroidal labeling on C2m × C2n is both row-sum balanced and column-sum balanced. In addition, when m = n is even, an antipodal balanced C4-face-magic toroidal labeling on C2n × C2n is diagonal-sum balanced.