Proof, that in any vertex coloring of graph $G$ using $\chi(G)$ coloros, there is a vertex of every color such that it has a neighbour in every color.
My idea is: let's assume that there is a color, which doesn't have a vertex with neighbour in every color. So let's change his color to a new, legal one and continue changing colors of other vertexes of the original color. This way we would remove one color, and thus color the graph using $\chi(G)-1$ colors, which is impossible.
It is important to notice, that changing the colors of previous vertex won't influence changing color of future vertex, as they can't be connected.