On directed versions of the Hajnal--Szemerédi theorem

We say that a (di)graph $G$ has a perfect $H$-packing if there exists a set of vertex-disjoint copies of $H$ which cover all the vertices in $G$. The seminal Hajnal--Szemerédi theorem characterises the minimum degree that ensures a graph $G$ contains a perfect $K_r$-packing. In this paper we prove the following analogue for directed graphs: Suppose that $T$ is a tournament on $r$ vertices and $G$ is a digraph of sufficiently large order $n$ where $r$ divides $n$. If $G$ has minimum in- and outdegree at least $(1-1/r)n$ then $G$ contains a perfect $T$-packing. In the case when $T$ is a cyclic triangle, this result verifies a recent conjecture of Czygrinow, Kierstead and Molla (for large digraphs). Furthermore, in the case when $T$ is transitive we conjecture that it suffices for every vertex in $G$ to have sufficiently large indegree or outdegree. We prove this conjecture for transitive triangles and asymptotically for all $r \geq 3$. Our approach makes use of a result of Keevash and Mycroft concerning almost perfect matchings in hypergraphs as well as the Directed Graph Removal lemma.

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