Decomposing complete 3-uniform hypergraph K_{n}^{(3)} into 7-cycles
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...
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AGH Univeristy of Science and Technology Press
2019-01-01
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| Series: | Opuscula Mathematica |
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| Online Access: | https://www.opuscula.agh.edu.pl/vol39/3/art/opuscula_math_3923.pdf |
| _version_ | 1828471046999638016 |
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| author | Meihua Meiling Guan Jirimutu |
| author_facet | Meihua Meiling Guan Jirimutu |
| author_sort | Meihua |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-12-11T05:04:25Z |
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| id | doaj.art-df1643f71bd645b5bb2ecf8b0f47fb9a |
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| language | English |
| last_indexed | 2024-12-11T05:04:25Z |
| publishDate | 2019-01-01 |
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| spelling | doaj.art-df1643f71bd645b5bb2ecf8b0f47fb9a2022-12-22T01:20:04ZengAGH Univeristy of Science and Technology PressOpuscula Mathematica1232-92742019-01-01393383393https://doi.org/10.7494/OpMath.2019.39.3.3833923Decomposing complete 3-uniform hypergraph K_{n}^{(3)} into 7-cyclesMeihua0Meiling Guan1Jirimutu2Mongolia University for the Nationalities, College of Mathematics of Inner, Tongliao, China 028043Mongolia University for the Nationalities, College of Mathematics of Inner, Tongliao, China 028043Mongolia University for the Nationalities, College of Mathematics of Inner, Tongliao, China 028043We 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.https://www.opuscula.agh.edu.pl/vol39/3/art/opuscula_math_3923.pdfuniform hypergraph7-cyclecycle decomposition |
| spellingShingle | Meihua Meiling Guan Jirimutu Decomposing complete 3-uniform hypergraph K_{n}^{(3)} into 7-cycles Opuscula Mathematica uniform hypergraph 7-cycle cycle decomposition |
| title | Decomposing complete 3-uniform hypergraph K_{n}^{(3)} into 7-cycles |
| title_full | Decomposing complete 3-uniform hypergraph K_{n}^{(3)} into 7-cycles |
| title_fullStr | Decomposing complete 3-uniform hypergraph K_{n}^{(3)} into 7-cycles |
| title_full_unstemmed | Decomposing complete 3-uniform hypergraph K_{n}^{(3)} into 7-cycles |
| title_short | Decomposing complete 3-uniform hypergraph K_{n}^{(3)} into 7-cycles |
| title_sort | decomposing complete 3 uniform hypergraph k n 3 into 7 cycles |
| topic | uniform hypergraph 7-cycle cycle decomposition |
| url | https://www.opuscula.agh.edu.pl/vol39/3/art/opuscula_math_3923.pdf |
| work_keys_str_mv | AT meihua decomposingcomplete3uniformhypergraphkn3into7cycles AT meilingguan decomposingcomplete3uniformhypergraphkn3into7cycles AT jirimutu decomposingcomplete3uniformhypergraphkn3into7cycles |