$C_4$-Face-Magic Torus Labelings on $C_4\times C_4$

Abstract

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 $f:V(G)\to \{1,2,\dots ,|V(G)|\}$ 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.