D-Distance magic labeling of $C^n_r$

Articles in Press

Document Type : Original paper

Authors

1 Birla Institute of Technology and Science Pilani, K K Birla Goa Campus, Goa, India

2 Rosary College of Commerce and Arts, Navelim, Salcete, Goa

Abstract

Let $G=(V, E)$ be a graph of order $n$. Let $D\subseteq\{0,1,2,\dots, \text{diam}(G)\}$ be nonempty. The $D$-neighborhood $N_D(x)$, of a vertex $x$ is the set of all vertices whose distance from vertex $x$ is an element in $D$, that is, $N_D(x)=\{y\in V:\ d(x,y)=m, m\in D\}$. A $D$-distance magic labeling of $G$ is a bijection $f\colon V\to \{1,2,\dots,n\}$ for which there exists a positive integer $k$, such that $\sum_{x\in N_D(v)}f(x)=k$ for all $v\in V$, where $N_D(v)$ is the $D$-open neighborhood of $v$. Let $\Gamma$ be an abelian group of order $n$. A $(\Gamma,D)$-distance magic labeling of $G$ is a bijection $l\colon V\to \Gamma$ for which there exists an element $\mu\in \Gamma$, such that $\sum_{x\in N_D(v)}l(x)=\mu$ for all $v\in V$. This paper presents the necessary and sufficient conditions for the existence of $D$-distance magic labeling for $C_n^r$ for a set $D$ containing elements in arithmetic progression. For the same set $D$, we also study the $(\Gamma, D)$-distance magic labeling of $C_n^r$ for some specific classes of abelian groups $\Gamma$.

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References

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Communications in Combinatorics and Optimization

Articles in Press, Accepted Manuscript
Available Online from 02 December 2025

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