Group distance magic graphs $G\times C_n$

TL;DR

This paper investigates conditions under which the direct product of a graph with a cycle admits a group distance magic labeling, providing new existence results based on graph order and degree congruences.

Contribution

It establishes new sufficient conditions for the existence of group distance magic labelings in graph products involving cycles, extending previous results.

Findings

Existence of $ ext{Gamma}$-distance magic labelings for $G imes C_4$ when $|V(G)|=2^p(2k+1)$

Existence of such labelings for $G imes C_8$ when the degree condition is even

Conditions depend on the order of the graph and degree congruences modulo powers of two

Abstract

A $\Gamma$-distance magic labeling of a graph $G=(V,E)$ with $|V | = n$ is a bijection $f$ from $V$ to an Abelian group $\Gamma$ of order $n$ such that the weight $w(x)=\sum_{y\in N_G(x)}f(y)$ of every vertex $x \in V$ is equal to the same element $\mu \in \Gamma$, called the \emph{magic constant}. In this paper we will show that if $G$ is a graph of order $n=2^{p}(2k+1)$ for some natural numbers $p$, $k$ such that $\deg(v)\equiv c \imod {2^{p+2}}$ for some constant $c$ for any $v\in V(G)$, then there exists a $\Gamma$-distance magic labeling for any Abelian group $\Gamma$ of order $4n$ for the direct product $G\times C_4$. Moreover if $c$ is even then there exists a $\Gamma$-distance magic labeling for any Abelian group $\Gamma$ of order $8n$ for the direct product $G\times C_8$.