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

Meihua
Meiling Guan
Jirimutu

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.
• Revised: 2018-08-22.
• Accepted: 2018-08-22.
• Published online: 2019-02-23.