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.