An Extension of the Hajnal–Szemerédi Theorem to Directed Graphs

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( $\vv G$ )= min v∈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.