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Distance magic (\(r, t\))-hypercycles. (English) Zbl 1365.05213
A distance magic labeling of a hypergraph \(H=(V,E)\) of order \(n\) is a bijection \(\ell:V \rightarrow [n]\) such that \(\sum_{x\in N(v)} \ell(x) = k\) holds for all \(v\in V\) and some positive integer \(k\). The \((r,t)\)-hypercycle, \(r\in [t-1]\), is a \(t\)-uniform hypergraph whose vertices can be ordered cyclically in such a way that the edges are segments of that cyclic order and every two consecutive edges share exactly \(r\) vertices. In this paper, the author proves that if \(t \in \{3,4,6\}\), then an \((r,t)\)-hypercycle of order \(n\) is distance magic if and only if \(r = t - 1\) and one of the following condition holds: (i) \(r = 2\) and \(n = 6\), (ii) \(r = 3\) and \(n = 8\) or \(n = 24\), (iii) \(r = 5\) and \(n \in \{12, 20, 60\}\). To obtain the result, distance magic labelings of powers of (graph) cycles are investigated first.

MSC:
05C65 Hypergraphs
05C78 Graph labelling (graceful graphs, bandwidth, etc.)
05C38 Paths and cycles
05C12 Distance in graphs
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