Let $G = (V, E) $ be a simple k-chromatic graph. A coloring of $G$ can be assumed to imply a proper k-coloring on the vertices.
We call a set of vertices $V'\subset V$ fully chromatic if for any coloring of $G$, all colors are present in $V'$. If $V'$ is minimal, in the sense that all subsets are not fully chromatic, then we call $V'$ critically fully chromatic.
Now I believe that a k-chromatic graph can't have a critically fully chromatic subset of size $k+1$. In the case k=4, this would simplify the proof of the 4CT by a lot.
Can some come up with such a critical chromatic set of vertices of size $k+1$?