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 on 3k vertices with minimum total degree δt( − → G) := minv∈V (deg −(v)+deg+(v)) ≥ 3(3k−1)/2 has a − → C 3-factor, where − → C 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 on 3k vertices with minimum total degree δt( − → G) ≥ 4k − 1 has a factor consisting of a copies of − → T 3 and b copies of − → C 3, where − → T 3 is the transitive tournament on three vertices. In particular, using b = 0, there is a − → T 3-factor of − → G , and using a = 1, it is possible to obtain a − → C 3-factor of − → G by reversing just one edge of − → G . All these results are phrased and proved more generally in terms of undirected multigraphs. We conjecture that every directed graph − → G on 3k vertices with minimum semidegree δ0( − → G) := minv∈V min(deg −(v), deg+(v)) ≥ 2k has a − → C 3-factor, and prove that this is asymptotically correct.