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arXiv · 2506.02455

Perfect $1$-factorisations of $K_{11,11}$

Jack Allsop·Ian M. Wanless
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Abstract

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.

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Jack Allsop, Ian M. Wanless. 2025-06-03. Perfect $1$-factorisations of $K_{11,11}$. https://arxiv.org/abs/2506.02455

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