On directed versions of the Corrádi–Hajnal corollary

Abstract

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 G⃗ on 3k vertices with minimum total degree δt(G⃗)≔minv∈V(deg−(v)+deg+(v))≥3(3k−1)/2 has a C⃗3-factor, where C⃗3 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 G⃗ on 3k vertices with minimum total degree δt(G⃗)≥4k−1 has a factor consisting of a copies of T⃗3 and b copies of C⃗3, where T⃗3 is the transitive tournament on three vertices. In particular, using b=0, there is a T⃗3-factor of G⃗, and using a=1, it is possible to obtain a C⃗3-factor of G⃗ by reversing just one edge of G⃗. All these results are phrased and proved more generally in terms of undirected multigraphs.We conjecture that every directed graph G⃗ on 3k vertices with minimum semidegree δ0(G⃗)≔minv∈Vmin(deg−(v),deg+(v))≥2k has a C⃗3-factor, and prove that this is asymptotically correct.