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Note on group distance magic complete bipartite graphs. (English) Zbl 1284.05122
Summary: A \(\Gamma\)-distance magic labeling of a graph \(G=(V,E)\) with \(| V|=n\) is a bijection \(\ell\) from \(V\) to an abelian group \(\Gamma\) of order \(n\) such that the weight \(w(x)=\sum_{y\in N_G(x)}\ell (y)\) of every vertex \(x\in V\) is equal to the same element \(\mu\in\Gamma\), called the magic constant. A graph \(G\) is called a group distance magic graph if there exists a \(\Gamma\)-distance magic labeling for every abelian group \(\Gamma\) of order \(| V(G)|\).
In this paper we give necessary and sufficient conditions for complete \(k\)-partite graphs of odd order \(p\) to be \(\mathbb Z_p\)-distance magic. Moreover we show that if \(p\equiv 2\) \(\pmod 4\) and \(k\) is even, then there does not exist a group \(\Gamma\) of order \(p\) such that there exists a \(\Gamma\)-distance labeling for a \(k\)-partite complete graph of order \(p\). We also prove that \(K_{m,n}\) is a group distance magic graph if and only if \(n+m\not\equiv 2\) \(\pmod 4\).

MSC:
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
05C78 Graph labelling (graceful graphs, bandwidth, etc.)
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