Abstract
A group distance magic labeling or a \({\mathcal{G}}\) -distance magic labeling of a graph G = (V, E) with \({|V | = n}\) is a bijection f from V to an Abelian group \({\mathcal{G}}\) 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 \({\mu \in \mathcal{G}}\) , called the magic constant. In this paper we will show that if G is a graph of order n = 2p(2k + 1) for some natural numbers p, k such that \({\deg(v)\equiv c \mod {2^{p+1}}}\) for some constant c for any \({v \in V(G)}\) , then there exists a \({\mathcal{G}}\) -distance magic labeling for any Abelian group \({\mathcal{G}}\) of order 4n for the composition G[C 4]. Moreover we prove that if \({\mathcal{G}}\) is an arbitrary Abelian group of order 4n such that \({\mathcal{G} \cong \mathbb{Z}_2 \times\mathbb{Z}_2 \times \mathcal{A}}\) for some Abelian group \({\mathcal{A}}\) of order n, then there exists a \({\mathcal{G}}\) -distance magic labeling for any graph G[C 4], where G is a graph of order n and n is an arbitrary natural number.
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The work was partially supported by National Science Centre grant nr 2011/01/D/ST/04104, as well as by the Polish Ministry of Science and Higher Education.
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Cichacz, S. Note on Group Distance Magic Graphs G[C 4]. Graphs and Combinatorics 30, 565–571 (2014). https://doi.org/10.1007/s00373-013-1294-z
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DOI: https://doi.org/10.1007/s00373-013-1294-z
Keywords
- Distance magic labeling
- Magic constant
- Sigma labeling
- Graph labeling
- Abelian group
- Composition of graphs
- Lexicographic product of graphs