Opuscula Math. 39, no. 3 (2019), 383-393
https://doi.org/10.7494/OpMath.2019.39.3.383
Opuscula Mathematica
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 \(n\leq17\), 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.
Keywords: uniform hypergraph, 7-cycle, cycle decomposition.
Mathematics Subject Classification: 05C65, 05C85.
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- Meihua
- Mongolia University for the Nationalities, College of Mathematics of Inner, Tongliao, China 028043
- Meiling Guan
- Mongolia University for the Nationalities, College of Mathematics of Inner, Tongliao, China 028043
- Jirimutu
- Mongolia University for the Nationalities, College of Mathematics of Inner, Tongliao, China 028043
- Communicated by Gyula O.H. Katona.
- Received: 2018-01-29.
- Revised: 2018-08-22.
- Accepted: 2018-08-22.
- Published online: 2019-02-23.
