On directed versions of the Corrádi-Hajnal corollary

For k∈N, Corrádi and Hajnal proved that every graph G on 3k vertices with minimum degree δ(G)≥2k has a C₃-factor, i.e.,a partitioning of the vertex set so that each part induces the 3-cycle C₃. Wang proved that every directed graph G{combining right arrow above} on 3k vertices with minimum total degree δt(G{combining right arrow above}){colon equals}minv∈V(deg-(v)+deg+(v))≥3(3k-1)/2 has a C{combining right arrow above}3-factor, where C{combining right arrow above}3 is the directed 3-cycle. The degree bound in Wang's result is tight. However, our main result implies that for all integers a≥1 and b≥0 with a+b=k, every directed graph G{combining right arrow above} on 3k vertices with minimum total degree δt(G{combining right arrow above})≥4k-1 has a factor consisting of a copies of T{combining right arrow above}3 and b copies of C{combining right arrow above}3, where T{combining right arrow above}3 is the transitive tournament on three vertices. In particular, using b=0, there is a T{combining right arrow above}3-factor of G{combining right arrow above}, and using a=1, it is possible to obtain a C{combining right arrow above}3-factor of G{combining right arrow above} by reversing just one edge of G{combining right arrow above}. All these results are phrased and proved more generally in terms of undirected multigraphs. We conjecture that every directed graph G{combining right arrow above} on 3. k vertices with minimum semidegree δ0(G{combining right arrow above}){colon equals}minv∈Vmin(deg-(v),deg+(v))≥2k has a C{combining right arrow above}3-factor, and prove that this is asymptotically correct.