is a selection of $n$ cells of the square, choosing exactly one from each row and column, such that every number in $[n]$ is contained in exactly one cell of the selection. Similarly, a *partial transversal* of a Latin square is a maximal selection of cells, each cell again being from a different row and column, such that no two chosen cells contain the same symbol.

As shown by Maillet (1894), there are many Latin squares which do not have complete transversals. However, a famous conjecture of Ryser (1967) states that every Latin square of odd order has a transversal and, moreover, Brualdi conjectured that every Latin square of order $n$ admits a partial transversal of size at least $n-1$.

Proofs for both conjectures seem out of reach, even though it is known that in every Latin square of order $n$, there are partial transversals of size $n-o(n)$. All this and more can be found in [@Denes].

How do Latin squares relate to rainbow cycles, or graph theory in general? Consider any Latin square $A$, and let $R$ and $C$ denote the sets of its rows and columns respectively. Then $A$ defines a proper edge coloring of the complete bipartite 