  K\"uhn, Osthus, and Townsend asked whether there exists a constant $C$ such
that every strongly $Ct$-connected tournament contains all possible $1$-factors
with at most $t$ components. We answer this question in the affirmative. This
is best possible up to constant. In addition, we can ensure that each cycle in
the $1$-factor contains a prescribed vertex.
  Indeed, we derive this result from a more general result on partitioning
digraphs which are close to semicomplete. More precisely, we prove that there
exists a constant $C$ such that for any $k\geq 1$, if a strongly
$Ck^4t$-connected digraph $D$ is close to semicomplete, then we can partition
$D$ into $t$ strongly $k$-connected subgraphs with prescribed sizes, provided
that the prescribed sizes are $\Omega(n)$. This result improves the earlier
result of K\"uhn, Osthus, and Townsend. Here, the condition of connectivity
being linear in $t$ is best possible, and the condition of prescribed size
being $\Omega(n)$ is also best possible.

Combinatorics