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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.

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