TITLE: Majority coloring for directed graphs
QUESTION [9 upvotes]: 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: 

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

REPLY [5 votes]: The paper mentioned above has been published:
Stephan Kreutzer, Sang-il Oum, Paul Seymour, Dominic van der Zypen, David R. Wood. 
"Majority Colourings of Digraphs"
Electronic Journal of Combinatorics, 24(2):#P2.25, 2017. 
http://www.combinatorics.org/ojs/index.php/eljc/article/view/v24i2p25
Other papers have appeared on this topic: 
Marcin Anholcer, Bartłomiej Bosek, Jarosław Grytczuk. 
"Every digraph is majority 4-choosable"
http://arxiv.org/abs/1608.06912
Fiachra Knox, Robert Šámal. 
"Linear Bound for Majority Colourings of Digraphs"
https://arxiv.org/abs/1701.05715
A. Girao, T. Kittipassorn, and K. Popielarz. 
"Generalised majority colourings of digraphs"
https://arxiv.org/abs/1701.03780