Perfect codes in \(m\)-Cayley hypergraphs. (English) Zbl 07919056
Summary: Let \(G\) be a finite group with the identity \(e\) and \(S\) be a subset of \(G \smallsetminus \{e \}\) such that \(S = S^{- 1}\). Let \(m\) be an integer such that \(2 \leq m \leq \max \{ o ( s ) \mid s \in S \}\), where \(o ( s )\) is the order of \(s\) in \(G\). The \(m\)-Cayley hypergraph \(\mathcal{H}\) of \(G\) over \(S, m - \operatorname{Cay} ( G , S )\), is a hypergraph with vertex set \(G\) and an edge set \(\{ \{ s^i x \mid 0 \leq i \leq m - 1 \} \mid x \in G , s \in S \} \). We discover a necessary and sufficient condition for a subset \(S\) of \(G \smallsetminus \{ e \}\) such that a subgroup \(H\) is a perfect code in \(m - \operatorname{Cay} ( G , S )\) and obtain some conditions for a subgroup \(H\) that guarantee the existence of a subset \(S \subseteq G \smallsetminus \{ e \}\) such that \(H\) is a perfect code in \(m - \operatorname{Cay} ( G , S )\).
MSC:
| 05C25 | Graphs and abstract algebra (groups, rings, fields, etc.) |
| 05C65 | Hypergraphs |
| 05C69 | Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.) |
| 94B25 | Combinatorial codes |
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