# ALGEBRAIC COMBINATORICS

Parity of transversals of Latin squares
Algebraic Combinatorics, Volume 3 (2020) no. 2, pp. 539-557.

We introduce a notion of parity for transversals, and use it to show that in Latin squares of order $2\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}4$, the number of transversals is a multiple of 4. We also demonstrate a number of relationships (mostly congruences modulo 4) involving ${E}_{1},\cdots ,{E}_{n}$, where ${E}_{i}$ is the number of diagonals of a given Latin square that contain exactly $i$ different symbols.

Let $A\left(i\mid j\right)$ denote the matrix obtained by deleting row $i$ and column $j$ from a parent matrix $A$. Define ${t}_{ij}$ to be the number of transversals in $L\left(i\mid j\right)$, for some fixed Latin square $L$. We show that ${t}_{ab}\equiv {t}_{cd}\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}2$ for all $a,b,c,d$ and $L$. Also, if $L$ has odd order then the number of transversals of $L$ equals ${t}_{ab}$ mod 2. We conjecture that ${t}_{ac}+{t}_{bc}+{t}_{ad}+{t}_{bd}\equiv 0\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}4$ for all $a,b,c,d$.

In the course of our investigations we prove several results that could be of interest in other contexts. For example, we show that the number of perfect matchings in a $k$-regular bipartite graph on $2n$ vertices is divisible by $4$ when $n$ is odd and $k\equiv 0\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}4$. We also show that

 $\mathrm{per}A\left(a\mid c\right)+\mathrm{per}A\left(b\mid c\right)+\mathrm{per}A\left(a\mid d\right)+\mathrm{per}A\left(b\mid d\right)\equiv 0\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}4$

for all $a,b,c,d$, when $A$ is an integer matrix of odd order with all row and columns sums equal to $k\equiv 2\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}4$.

Revised:
Accepted:
Published online:
DOI: https://doi.org/10.5802/alco.103
Classification: 05B15,  15A15,  05C70
Keywords: parity, Latin square, transversal, permanent, Latin rectangle, perfect matching, permanental minor, bipartite graph
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author = {Best, Darcy and Wanless, Ian M.},
title = {Parity of transversals of {Latin} squares},
journal = {Algebraic Combinatorics},
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Best, Darcy; Wanless, Ian M. Parity of transversals of Latin squares. Algebraic Combinatorics, Volume 3 (2020) no. 2, pp. 539-557. doi : 10.5802/alco.103. https://alco.centre-mersenne.org/articles/10.5802/alco.103/

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