Tight 9-Cycle Decompositions of λ -Fold Complete 3-Uniform Hypergraphs

Hongtao Zhao and Jianxiao Gu ()
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Hongtao Zhao: School of Mathematics and Physics, North China Electric Power University, Beijing 102206, China
Jianxiao Gu: School of Mathematics and Physics, North China Electric Power University, Beijing 102206, China

Mathematics, 2024, vol. 12, issue 19, 1-16

Abstract: For 2 ≤ t ≤ m , let Z m denote the group of integers modulo m , and let T C m ( t ) denote the t -uniform hypergraph with vertex set Z m and hyperedge set { { i , i + 1 , i + 2 , … , i + t − 1 } : i ∈ Z m } . Any hypergraph isomorphic to T C m ( t ) is a t -uniform tight m -cycle. In this paper, we consider the existence of tight 9-cycle decompositions of λ -fold complete 3-uniform hypergraphs. According to the recursive constructions, the required designs of small orders are found. For hypergraphs with large orders, they can be recursively generated using some designs of small orders. Then, we obtain the necessary and sufficient conditions for the existence of T C 9 ( 3 ) -decomposition of λ K n ( 3 ) . We show there exists a T C 9 ( 3 ) -decomposition of λ K n ( 3 ) if and only if λ n ( n − 1 ) ( n − 2 ) ≡ 0 ( mod 54 ) , λ ( n − 1 ) ( n − 2 ) ≡ 0 ( mod 6 ) and n ≥ 9 .

Keywords: hypergraph decomposition; tight cycle; ( H ,?)-design; complete 3-uniform hypergraph (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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