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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| Accès en ligne: | https://arxiv.org/abs/2402.10081 |
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| _version_ | 1866929244418867200 |
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| author | Huerta, Zazil Santizo Keranen, Melissa |
| author_facet | Huerta, Zazil Santizo Keranen, Melissa |
| contents | The uniform Hamilton-Waterloo Problem (HWP) asks for a resolvable $(C_M, C_N)$-decomposition of $K_v$ into $α$ $C_M$-factors and $β$ $C_N$-factors. We denote a solution to the uniform Hamilton Hamilton-Waterloo problem by $\hbox{HWP}(v; M, N; α, β)$. Our research concentrates on addressing some of the remaining unresolved cases, which pose a significant challenge to generalize. We place a particular emphasis on instances where the $\gcd(M,N)=\{2, 3\}$, with a specific focus on the parameter $M=6$. We introduce modifications to some known structures, and develop new approaches to resolving these outstanding challenges in the construction of uniform $2$-factorizations. This innovative method not only extends the scope of solved cases, but also contributes to a deeper understanding of the complexity involved in solving the Hamilton-Waterloo Problem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_10081 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Hamilton-Waterloo Problem with a single factor of 6-cycles Huerta, Zazil Santizo Keranen, Melissa Combinatorics The uniform Hamilton-Waterloo Problem (HWP) asks for a resolvable $(C_M, C_N)$-decomposition of $K_v$ into $α$ $C_M$-factors and $β$ $C_N$-factors. We denote a solution to the uniform Hamilton Hamilton-Waterloo problem by $\hbox{HWP}(v; M, N; α, β)$. Our research concentrates on addressing some of the remaining unresolved cases, which pose a significant challenge to generalize. We place a particular emphasis on instances where the $\gcd(M,N)=\{2, 3\}$, with a specific focus on the parameter $M=6$. We introduce modifications to some known structures, and develop new approaches to resolving these outstanding challenges in the construction of uniform $2$-factorizations. This innovative method not only extends the scope of solved cases, but also contributes to a deeper understanding of the complexity involved in solving the Hamilton-Waterloo Problem. |
| title | On the Hamilton-Waterloo Problem with a single factor of 6-cycles |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2402.10081 |