# C_4-face-magic toroidal labelings on C_m × C_n

### Abstract

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 *C*_{n}-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* ≅ *C*_{n}, the sum of all the vertex labels along *C*_{n} is a constant *S*. Let *x*_{v} = *f*(*v*) for all *v* ∈ *V*(*G*). We call {*x*_{v} : *v* ∈ *V*(*G*)} a *C*_{n}-face-magic toroidal labeling on *G*. We show that, for all *m*, *n* ≥ 2, *C*_{m} × *C*_{n} admits a *C*_{4}-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 *C*_{4}-face-magic toroidal labeling {*x*_{i, j} : (*i*, *j*) ∈ *V*(*C*_{2m} × *C*_{2n})} on *C*_{2m} × *C*_{2n} is antipodal balanced if $x_{i,j} + x_{i+m,j+n} = \tfrac{1}{2} S$, for all (*i*, *j*) ∈ *V*(*C*_{2m} × *C*_{2n}). We show that there exists an antipodal balanced *C*_{4}-face-magic toroidal labeling on *C*_{2m} × *C*_{2n} if and only if the parity of *m* and *n* are the same. Furthermore, when both *m* and *n* are even, an antipodal balanced *C*_{4}-face-magic toroidal labeling on *C*_{2m} × *C*_{2n} is both row-sum balanced and column-sum balanced. In addition, when *m* = *n* is even, an antipodal balanced *C*_{4}-face-magic toroidal labeling on *C*_{2n} × *C*_{2n} is diagonal-sum balanced.