Spectrum of 3-uniform 6- and 9-cycle systems over K (3) v − I

Keszler, Anita and Tuza, Zsolt (2024) Spectrum of 3-uniform 6- and 9-cycle systems over K (3) v − I. DISCRETE MATHEMATICS, 347 (3). ISSN 0012-365X

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Abstract

We consider edge decompositions of K (3) v − 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 (3) v.

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