Perfect $1$-factorisations of $K_{11,11}$
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866911577690603520 |
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| author | Allsop, Jack Wanless, Ian M. |
| author_facet | Allsop, Jack Wanless, Ian M. |
| contents | A perfect $1$-factorisation of a graph is a decomposition of that graph into $1$-factors such that the union of any two $1$-factors is a Hamiltonian cycle. A Latin square of order $n$ is row-Hamiltonian if for every pair $(r,s)$ of distinct rows, the permutation mapping $r$ to $s$ has a single cycle of length $n$. We report the results of a computer enumeration of the perfect $1$-factorisations of the complete bipartite graph $K_{11,11}$. This also allows us to find all row-Hamiltonian Latin squares of order $11$. Finally, we plug a gap in the literature regarding how many row-Hamiltonian Latin squares are associated with the classical families of perfect $1$-factorisations of complete graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_02455 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Perfect $1$-factorisations of $K_{11,11}$ Allsop, Jack Wanless, Ian M. Combinatorics 05C70, 05B15 A perfect $1$-factorisation of a graph is a decomposition of that graph into $1$-factors such that the union of any two $1$-factors is a Hamiltonian cycle. A Latin square of order $n$ is row-Hamiltonian if for every pair $(r,s)$ of distinct rows, the permutation mapping $r$ to $s$ has a single cycle of length $n$. We report the results of a computer enumeration of the perfect $1$-factorisations of the complete bipartite graph $K_{11,11}$. This also allows us to find all row-Hamiltonian Latin squares of order $11$. Finally, we plug a gap in the literature regarding how many row-Hamiltonian Latin squares are associated with the classical families of perfect $1$-factorisations of complete graphs. |
| title | Perfect $1$-factorisations of $K_{11,11}$ |
| topic | Combinatorics 05C70, 05B15 |
| url | https://arxiv.org/abs/2506.02455 |