Note on Group Distance Magic Graphs G[C 4] |
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Authors: | Sylwia Cichacz |
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Affiliation: | 1. Faculty of Applied Mathematics, AGH University of Science and Technology, Al. Mickiewicza 30, 30-059, Kraków, Poland
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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 ${w(x) = sum_{yin N_G(x)}f(y)}$ 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 = 2 p (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 ${mathcal{G} cong mathbb{Z}_2 timesmathbb{Z}_2 times mathcal{A}}$ 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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