\(C_{4}\)-Face-Magic Torus Labelings on \(C_{4} \times C_{4}\)

For a toroidal graph \( G = (V, E) \) embedded in the torus, let \( \mathcal{F}(G) \) denote the set of faces of \( G \). Then, \( G \) is called a \( C_{n} \)-face-magic torus graph if there exists a bijection \( f: V(G) \rightarrow \{1, 2, \ldots, |V(G)|\} \) such that for any \( F \in \mathcal{F}(G) \) with \( F \cong C_{n} \), the sum of all the vertex labelings along \( C_{n} \) is a constant \( S \).

Let \( x_{v} = f(v) \) for all \( v \in V(G) \). We call \( \{x_{v} : v \in V(G)\} \) a \( C_{n} \)-face magic torus labeling on \( G \).

We say that a \( C_{4} \)-face-magic torus labeling \( \{x_{i,j} \} \) on \( C_{2n} \times C_{2n} \) is antipodal balanced if \( x_{i,j} + x_{i+n,j+n} = \frac{1}{2}S \) for all \( (i, j) \in V(C_{2n} \times C_{2n}) \).

We determine all antipodal balanced \( C_{4} \)-face-magic torus labelings on \( C_{4} \times C_{4} \) up to symmetries on a torus.