On directed versions of the Corr\'adi-Hajnal Corollary

Abstract

For $k \in \mathbb N$, Corr\'adi and Hajnal proved that every graph $G$ on $3k$ vertices with minimum degree $\delta(G) \ge 2k$ 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 $\overrightarrow G$ on $3k$ vertices with minimum total degree $\delta_t(\overrightarrow G):=\min_{v\in V}(deg^-(v)+deg^+(v)) \ge 3(3k-1)/2$ has a $\overrightarrow C_3$-factor, where $\overrightarrow C_3$ is the directed 3-cycle. The degree bound in Wang's result is tight. However, our main result implies that for all integers $a \ge 1$ and $b \ge 0$ with $a+b=k$, every directed graph $\overrightarrow G$ on $3k$ vertices with minimum total degree $\delta_t(\overrightarrow G)\ge 4k-1$ has a factor consisting of $a$ copies of $\overrightarrow T_3$ and $b$ copies of $\overrightarrow C_3$, where $\overrightarrow T_3$ is the transitive tournament on three vertices. In particular, using $b=0$, there is a $\overrightarrow T_3$-factor of $\overrightarrow G $, and using $a=1$, it is possible to obtain a $\overrightarrow C_3$-factor of $\overrightarrow G$ by reversing just one edge of $\overrightarrow G$. All these results are phrased and proved more generally in terms of undirected multigraphs. We conjecture that every directed graph $\overrightarrow G$ on $3k$ vertices with minimum semidegree $\delta_0(\overrightarrow G):=\min_{v\in V}\min(deg^-(v),deg^+(v)) \ge 2k$ has a $\overrightarrow C_3$-factor, and prove that this is asymptotically correct.