Supermagic labeling of $C_n\Box C_m$

TL;DR

This paper proves Ivančo's conjecture that all Cartesian products of cycles with odd, non-coprime lengths admit a supermagic labeling, extending known results to a new class of graphs.

Contribution

The paper confirms Ivančo's conjecture for all cycle pairs with odd, non-coprime lengths, broadening the class of graphs known to have supermagic labelings.

Findings

Supermagic labelings exist for all $C_n \Box C_m$ with odd, non-coprime $n,m$.

Extends previous results to new cases of cycle product graphs.

Supports Ivančo's conjecture for a wider range of cycle pairs.

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.