graph with partite sets $C$ and $R$, as follows: for $c\in C$ and $r\in R$, the edge $\{c,r\}$ is colored with the number contained in the cell determined by the row $r$ and the column $c$. Then a transversal of $A$ corresponds to a perfect bipartite matching in this graph, in which no two edges use the same color: a rainbow perfect matching. Rainbow matchings are studied for example in [@Wang2008].

Conversely, every proper edge coloring of $K_n$ with $n-1$ colors can be used to construct a Latin square $A$ of order $n$, in the following way. Let $\{v_1,\dotsc,v_n\}$ denote the vertex set of $K_n$. Then for every $i\in [n]$ and $j\neq i$, let $A_{i,i}=n$ and let $A_{i,j}$ be the color of the edge $\{v_i,v_j\}$. A complete transversal of this Latin square corresponds to a 2-regular rainbow subgraph (i.e., a subgraph consisting of vertex-disjoint cycles) that covers all but at most one vertex of $K_n$. This can be seen as follows. For every vertex $v_i$, either $A_{i,i}$ is in the transversal, or two cells $A_{i,j}$ and $A_{k,i}$ (with $i\neq j$ and $i \neq k$) are in the transversal. In the former case, $v_i$ does not belong to the subgraph. In 