On directed versions of the Corrádi–Hajnal corollary

For k ∈ N , Corrádi and Hajnal proved that every graph G on 3 k vertices with minimum degree δ ( G ) ≥ 2 k has a C 3 -factor, i.e., a partitioning of the vertex set so that each part induces the 3-cycle C 3 . Wang proved that every directed graph G ⃗ on 3 k vertices with minimum total degree δ t ( G ⃗ ) ≔ min v ∈ V ( d e g − ( v ) + d e g + ( v ) ) ≥ 3 ( 3 k − 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 3 k vertices with minimum total degree δ t ( G ⃗ ) ≥ 4 k − 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. jecture that every directed graph G ⃗ on 3 k vertices with minimum semidegree δ 0 ( G ⃗ ) ≔ min v ∈ V min ( d e g − ( v ) , d e g + ( v ) ) ≥ 2 k has a C ⃗ 3 -factor, and prove that this is asymptotically correct.