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We consider edge decompositions of K v (3) − I , the complete 3-uniform hypergraph of order v minus a set of v / 3 mutually disjoint edges (1-factor). We prove that a decomposition into tight 6-cycles exists if and only if v ≡ 0 , 3 , 6 (mod 12) and v ≥ 6 ; and a decomposition into tight 9-cycles exists for all v ≥ 9 divisible by 3. These results are complementary to the theorems of Akin et al. [Discrete Math. 345 (2022)] and Bunge et al. [Australas. J. Combin. 80 (2021)] who settled the case of K v (3). [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of constructing hypercycle systems of 5-cycles in complete 3-uniform hypergraphs. A hypercycle system C (r , k , v) of order v is a collection of r-uniform k-cycles on a v-element vertex set, such that each r-element subset is an edge in precisely one of those k-cycles. We present cyclic hypercycle systems C (3 , 5 , v) of orders v = 25 , 26 , 31 , 35 , 37 , 41 , 46 , 47 , 55 , 56 , a highly symmetric construction for v = 40 , and cyclic 2-split constructions of orders 32 , 40 , 50 , 52 . As a consequence, all orders v ≤ 60 permitted by the divisibility conditions admit a C (3 , 5 , v) system. New recursive constructions are also introduced. [ABSTRACT FROM AUTHOR]