Decomposing complete 3-uniform hypergraph Kn(3) into 7-cycles

Abstract

We use the Katona-Kierstead definition of a Hamiltonian cycle in a uniform hypergraph. A decomposition of complete $k$-uniform hypergraph $K^{(k)}_{n}$ into Hamiltonian cycles was studied by Bailey-Stevens and Meszka-Rosa. For $n\equiv 2,4,5\pmod 6$, we design an algorithm for decomposing the complete 3-uniform hypergraphs into Hamiltonian cycles by using the method of edge-partition. A decomposition of $K^{(3)}_{n}$ into 5-cycles has been presented for all admissible $n \leq 17$, and for all $n=4^{m}+1$ when $m$ is a positive integer. In general, the existence of a decomposition into 5-cycles remains open. In this paper, we show if $42~|~(n-1)(n-2)$ and if there exist $\lambda=\frac{(n-1)(n-2)}{42}$ sequences $(k_{i_{0}},k_{i_{1}},\ldots,k_{i_{6}})$ on $D_{all}(n)$, then $K^{(3)}_{n}$ can be decomposed into 7-cycles. We use the method of edge-partition and cycle sequence. We find a decomposition of $K^{(3)}_{37}$ and $K^{(3)}_{43}$ into 7-cycles.