I came up with the following coloring concept when studying neural networks (which are often modelled using directed graphs). No idea whether there is already an established name for it.

If $X$ is a non-empty set, we say that $M\subseteq X$ is a *majority* if $|M| > |X\setminus M|$.

Let $G=(V,E)$ be a finite directed graph. For $v\in V$ we set $\text{In}(v)=\{x \in V: (x,v) \in E\}$.

Let $n$ be a positive integer. We say that a map $c:V(G) \to \{1,\ldots,n\}$ is a *majority coloring* if the following condition is satisfied:

> For every $v\in V(G)$ with $\text{In}(v) \neq \emptyset$, if for some $k \in \{1,\ldots, n\}$ we have that $c^{-1}(\{k\}) \cap \text{In}(v)$ is a majority of $\text{In}(v)$, then $c(v) \neq k$.

We set the *majority coloring number* $\chi_m(G)$ to be the least positive integer $j$ such that there is a majority coloring $c:V(G) \to \{1,\ldots,j\}$.

Many directed graphs I've looked at have majority coloring number $2$, but for instance $K_3$ with the orientation $1\to 2\to 3\to 1$ has majority coloring number $3$. 

**Questions**: 

  1. For $n\in\mathbb{N}$ is there a directed graph $G$ such that $\chi_m(G) = n$?
  2. For $n\in\mathbb{N}$ is there even a [tournament][1] $T$ such taht $\chi_m(T) = n$?


  [1]: https://en.wikipedia.org/wiki/Tournament_(graph_theory)