Spectrum of 3-uniform 6- and 9-cycle systems over [formula omitted]

Abstract

We consider edge decompositions of Kv(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 Kv(3).