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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 $F$ . 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 Häggkvist. If $G$ is a spanning subgraph of the blow-up of $C_k$ with parts of size $n$ and $\delta ^*(G) \ge \left(1 + \frac 1k\right)\frac n2 + 1$ , then $G$ contains $n$ vertex disjoint copies of $C_k$ such that each $C_k$ intersects each of the $k$ parts exactly once. A similar conjecture was also made by Fischer and the case $k=3$ was proved for large $n$ by Magyar and Martin.In this paper, we prove the conjecture of Häggkvist asymptotically. We also pose a conjecture which generalises this result by allowing the minimum degree conditions in each bipartite subgraph induced by pairs of parts of $G$ to vary. We support this new conjecture by proving the triangle case. This result generalises Johannson’s result asymptotically.

Hajnal and Szemerédi proved that every graph G with |G| = ks and δ(G)⩾ k(s − 1) contains k disjoint s-cliques; moreover this degree bound is optimal. We extend their theorem to directed graphs by showing that every directed graph $\vv G$ with |$\vv G$| = ks and δ($\vv G$) ⩾ 2k(s − 1) − 1 contains k disjoint transitive tournaments on s vertices, where δ($\vv G$)= minv∈V($\vv G$)d−(v)+d+(v). Our result implies the Hajnal–Szemerédi theorem, and its degree bound is optimal. We also make some conjectures regarding even more general results for multigraphs and partitioning into other tournaments. One of these conjectures is supported by an asymptotic result.