Hypergraphs provide a powerful framework for modeling polyadic relationships, generalizing classical graph structures. $ t $-Cayley hypergraphs, a natural extension of Cayley graphs, are inherently vertex-transitive under the automorphism group ($ \mathrm{Aut} $). However, this vertex-transitivity is not necessarily preserved when replacing $ \mathrm{Aut} $ with non-injective endomorphisms ($ \mathrm{End}' $). This work investigates the precise conditions under which $ t $-Cayley hypergraphs of finite cyclic groups retain this property. Our main result establishes that a $ t $-Cayley hypergraph $ H $ over the cyclic group $ \mathbb{Z}_n $ is $ \mathrm{End}' $-vertex-transitive if $ t $ divides $ n $.
Citation: Tanapat Chalarux, Sayan Panma. On $t$-Cayley hypergraphs of cyclic groups with vertex transitive property on the set of non-injective endomorphisms[J]. AIMS Mathematics, 2026, 11(6): 19031-19045. doi: 10.3934/math.2026775
Hypergraphs provide a powerful framework for modeling polyadic relationships, generalizing classical graph structures. $ t $-Cayley hypergraphs, a natural extension of Cayley graphs, are inherently vertex-transitive under the automorphism group ($ \mathrm{Aut} $). However, this vertex-transitivity is not necessarily preserved when replacing $ \mathrm{Aut} $ with non-injective endomorphisms ($ \mathrm{End}' $). This work investigates the precise conditions under which $ t $-Cayley hypergraphs of finite cyclic groups retain this property. Our main result establishes that a $ t $-Cayley hypergraph $ H $ over the cyclic group $ \mathbb{Z}_n $ is $ \mathrm{End}' $-vertex-transitive if $ t $ divides $ n $.
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Tanapat Chalarux, Sayan Panma. On $t$-Cayley hypergraphs of cyclic groups with vertex transitive property on the set of non-injective endomorphisms[J]. AIMS Mathematics, 2026, 11(6): 19031-19045. doi: 10.3934/math.2026775
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