Spectrum of $3$-uniform $6$- and $9$-cycle systems over $K_v^{(3)}-I$

TL;DR

This paper investigates the conditions under which complete 3-uniform hypergraphs minus a 1-factor can be decomposed into tight 6- and 9-cycles, establishing existence criteria based on the order of the hypergraph.

Contribution

It provides necessary and sufficient conditions for decomposing $K_v^{(3)}-I$ into tight 6- and 9-cycles, extending previous results in hypergraph cycle decompositions.

Findings

Decomposition into tight 6-cycles exists iff v ≡ 0,3,6 (mod 12) and v ≥ 6.

Decomposition into tight 9-cycles exists for all v ≥ 9 divisible by 3.

Results complement prior theorems by Akin et al. and Bunge et al.

Abstract

We consider edge decompositions of $K_v^{(3)}-I$, the complete 3-uniform hypergraph of order $v$ minus a 1-factor (parallel class, packing of $v/3$ disjoint edges). We prove that a decomposition into tight 6-cycles exists if and only if $v\equiv 0,3,6$ (mod 12) and $v\geq 6$; and a decomposition into tight 9-cycles exists for all $v\geq 9$ 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)].