I want to prove that for all $n$, there exists a constant $c(n)$ such that if $G=(V,E)$ is a directed graph with in-degree bounded by $n$, it is possible to partition the set of vertices $V$ to two subsets $V_1, V_2$ in a way that for all vertex $v$ in the graph induced by $V_1$ AND in the graph induced by $V_2$ , if the vertex $v$ is with out degree $d(v) \geq c(n)$ in $G$, it satisfies that its out degree in the corresponding subgraph will be between $\frac{1}{3}d(v)$ and $\frac{2}{3}d(v)$.
I suspect that it is the kind of statements which can be proved by the probabilistic method, using the local lemma (since there are some similar theorems proven in this way). However I couldn't figure out how to do that, will be glad for help.