Karp hardness of directed monochromatic triangle problem

Question

Monochromatic problem is a classic NP-complete problem.

Does the complexity stay NP-complete if we use directed graph?

DIRECTED MONOCHROMATIC TRIANGLE problem:

Input: A digraph $G(V,A)$

Output: YES if there exists a triangle-free bipartition of $A$, otherwise NO.

A triangle-free bipartition of $A$ is a partition $A=A_1\dot{\cup}A_2$ such that both $G[V,A_1]$ and $G[V,A_2]$ are triangle-free.

Note that a triangle in a digraph is the set of $3$ vertices $\{u,v,w\}$ such that $(u,v),(v,w),(w,u)\in A$. It means that we only allow triangle that has consecutive arcs. For example, $(u,v),(u,w),(w,u)$ does NOT form a triangle.

Answer 1

All directed graphs can be edge-partitioned into two subgraphs that are acyclic and therefore triangle free.

Let $\prec$ be any total ordering of the vertices. For each edge $(u, v) \in A$, put it in $A_1$ if $u \prec v$, and in $A_2$ otherwise. Both $(V, A_1)$ and $(V, A_2)$ must be acyclic.