  We define a cover of a Latin square to be a set of entries that includes at
least one representative of each row, column and symbol. A cover is minimal if
it does not contain any smaller cover. A partial transversal is a set of
entries that includes at most one representative of each row, column and
symbol. A partial transversal is maximal if it is not contained in any larger
partial transversal. We explore the relationship between covers and partial
transversals.
  We prove the following: (1) The minimum size of a cover in a Latin square of
order $n$ is $n+a$ if and only if the maximum size of a partial transversal is
either $n-2a$ or $n-2a+1$. (2) A minimal cover in a Latin square of order $n$
has size at most $\mu_n=3(n+1/2-\sqrt{n+1/4})$. (3) There are infinitely many
orders $n$ for which there exists a Latin square having a minimal cover of
every size from $n$ to $\mu_n$. (4) Every Latin square of order $n$ has a
minimal cover of a size which is asymptotically equal to $\mu_n$. (5) If $1\le
k\le n/2$ and $n\ge5$ then there is a Latin square of order $n$ with a maximal
partial transversal of size $n-k$. (6) For any $\epsilon>0$, asymptotically
almost all Latin squares have no maximal partial transversal of size less than
$n-n^{2/3+\epsilon}$.
