# C4-face-magic toroidal labelings on Cm × Cn

1-1-2021

Article

## Publication Title

Art of Discrete and Applied Mathematics

4

1

## DOI

10.26493/2590-9770.1368.f37

## 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 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 xi,j + xi+m,j+n = 21 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.

## Keywords

C4-face-magic graphs, Cartesian products of cycles, Polyomino, Toroidal graphs

This article originally appeared in The Art of Discrete and Applied Mathematics, Volume 4, No. 1, 2021. The article can also be found online at: