Note on group distance magic graphs $G[C_4]$

TL;DR

This paper investigates conditions under which certain graphs formed by replacing vertices with 4-cycles admit group distance magic labelings, expanding understanding of labelings in graph theory.

Contribution

It establishes new sufficient conditions for the existence of group distance magic labelings in graphs of the form G[C_4], including specific group structures and degree conditions.

Findings

Existence of group distance magic labelings for graphs with degrees congruent modulo 2^{p+1}.

Labelings are guaranteed for graphs G[C_4] with Abelian groups of order 4n of a specific form.

Provides new classes of graphs known to admit group distance magic labelings.

Abstract

A \emph{group distance magic labeling} or a $\gr$-distance magic labeling of a graph $G(V,E)$ with $|V | = n$ is an injection $f$ from $V$ to an Abelian group $\gr$ 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 $\mu \in \gr$, called the 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 $\deg(v)\equiv c \imod {2^{p+1}}$ for some constant $c$ for any $v\in V(G)$, then there exists an $\gr$-distance magic labeling for any Abelian group $\gr$ for the graph $G[C_4]$. Moreover we prove that if $\gr$ is an arbitrary Abelian group of order $4n$ such that $\gr \cong \zet_2 \times\zet_2 \times \gA$ for some Abelian group $\gA$ of order $n$, then exists a $\gr$-distance magic labeling for any graph $G[C_4]$.