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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