  For a subgraph $G$ of the blow-up of a graph $F$, we let $\delta^*(G)$ be the
smallest minimum degree over all of the bipartite subgraphs of $G$ induced by
pairs of parts that correspond to edges of $F$. In [Triangle-factors in a
balanced blown-up triangle. Discrete Mathematics, 2000], Johansson proved that
if $G$ is a spanning subgraph of the blow-up of $C_3$ with parts of size $n$
and $\delta^*(G) \ge \frac{2}{3}n + \sqrt{n}$, then $G$ contains $n$
vertex-disjoint triangles, and presented the following conjecture of
H\"aggkvist: If $G$ is a spanning subgraph of the blow-up of $C_k$ with parts
of size $n$ and $\delta^*(G) \ge (1 + 1/k)n/2 + 1$, then $G$ contains $n$
vertex disjoint copies of $C_k$ such that each $C_k$ intersects each of the $k$
parts exactly once. The degree condition of this conjecture is tight when $k=3$
and cannot be strengthened by more than one when $k \ge 4$., A similar
conjecture was also made by Fischer in [Variants of the Hajnal-Szemer\'edi
Theorem. Journal of Graph Theory, 1999] and the triangle case was proved for
large $n$ by Magyar and Martin in [Tripartite version of the Corr\'adi-Hajnal
Theorem. Discrete Mathematics, 2002].
  In this paper, we prove this Conjecture asymptotically. We also pose a
conjecture which generalizes this result by allowing the minimum degree
conditions on the nonempty bipartite subgraphs induced by pairs of parts to
vary. Our second result supports this new conjecture by proving the triangle
case. This result generalizes Johannson's result asymptotically.
