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arXiv · 2402.06205

Canonical labelling of Latin squares in average-case polynomial time

Michael J. Gill·Adam Mammoliti·Ian M. Wanless
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Abstract

A Latin square of order $n$ is an $n\times n$ matrix in which each row and column contains each of $n$ symbols exactly once. For $\epsilon>0$, we show that with high probability a uniformly random Latin square of order $n$ has no proper subsquare of order larger than $n^{1/2}\log^{1/2+\epsilon}n$. Using this fact we present a canonical labelling algorithm for Latin squares of order $n$ that runs in average time bounded by a polynomial in $n$. The algorithm can be used to solve isomorphism problems for many combinatorial objects that can be encoded using Latin squares, including quasigroups, Steiner triple systems, Mendelsohn triple systems, $1$-factorisations, nets, affine planes and projective planes.

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Michael J. Gill, Adam Mammoliti, Ian M. Wanless. 2024-02-09. Canonical labelling of Latin squares in average-case polynomial time. https://arxiv.org/abs/2402.06205

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