# *Friendly* coloring of a digraph

Given a simple, finite, directed graph $$G$$, let's call a $$2$$-coloring of the vertices of $$G$$ is friendly if every vertex has an out-neighbour of the same color (while not all vertices are same-colored).

Broadly speaking, I wonder what graphs admit a friendly coloring. To be more specific:

Is it true that every simple, finite, directed graph with the minimum outdegree $$3$$ has a friendly coloring? If this is false, does there exist an integer $$k\ge 4$$ such that every directed graph with the minimum outdegree $$k$$ has a friendly coloring? If so, what is the smallest $$k$$ with this property?

The case $$k=2$$ is easy, but already the case $$k=3$$ seems nontrivial to me.

The same questions can also be asked for three or more colors.

Carsten Thomassen (1983) proved that a digraph with minimal out-degree at least 3 has two vertex-disjoint cycles, call them $$C_1$$, $$C_2$$ and color black and white respectively. Then proceed as follows: on each step we have some vertices already colored, and from each colored vertex there exists an edge to a vertex of the same color. Take a non-colored vertex, and start a path from it until you get a cycle or go to an already colored vertex, say to a white one. Color all this path (or cycle) white. So finally you get a friendly coloring.