Paths and cycles of hypergraphs

“…Recently, in [5] there were given necessary and sufficent conditions for a decompostion of the complete 3-uniform hypergraph of order n into 4-cycles. Meanwhile, many papers studied the two different definitions of a Hamiltonian cycle in [6,13], which are due to Katona and Kierstead, Wang and Lee, respectively. In fact, the two different definitions of a Hamiltonian cycle are the same.…”

Section: Introductionmentioning

confidence: 99%

Decomposing complete 3-uniform hypergraph K_{n}^{(3)} into 7-cycles

Meihua¹,

Meiling²,

Jirimutu³

2019

Opuscula Math.

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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.

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“…Katona-Kierstead and Jianfang Wang gave the definition of Hamiltonian chain and Hamiltonian cycle in [1][2], independently. In fact, two different definitions of Hamiltonian chain and Hamiltonian cycle are the same.…”

Section: Introductionmentioning

confidence: 99%

“…In fact, two different definitions of Hamiltonian chain and Hamiltonian cycle are the same. Some researchers studied the decomposition of complete 3-uniform hypergraph   3 n K into Hamiltonian cycles and not Hamiltonian cycles in [2][3][4][5][6][7][8][9]. Especially , Bailey Stevens [3] used clique-finding the decomposition of   3 n K into Hamiltonian cycles for   3 7 K ,   3 8 K and Meszka-Rosa [4] showed that Hamiltonian decompositions of …”

Section: Introductionmentioning

confidence: 99%