Supermagic labeling of $C_n\Box C_m$

Abstract

A supermagic labeling (often also called supermagic labeling) of a graph $G(V,E)$ with $|E|=k$ is a bijection from $E$ to the set of first $k$ positive integers such that the sum of labels of all incident edges of every vertex $x\in V$ is equal to the same integer $c$. An existence of a supermagic labeling of Cartesian product of two cycles, $C_{n}\Box C_m$ for $n,m\geq4$ and both $n,m$ even and for any $C_n\Box C_n$ with $n\geq3$ was proved by Ivan\v{c}o. Ivan\v{c}o also conjectured that such labeling is possible for any $C_n\Box C_m$ with $n,m\geq3$. We prove his conjecture for all $n,m$ odd that are not relatively prime.