TITLE: Connection between chromatic number and independence number of a graph
QUESTION [6 upvotes]: Is it true that one can always colour a graph G with $\chi(G)$ colours in such a way that one of the colour classes is a maximum possible cardinality independent set? Please prove if it's true.  
[Note: $\chi(G)$ is the chromatic number of G, or the smallest number k such that we can colour the graph G with k colours. Independent set means none of the vertices in the set are adjacent.]

REPLY [7 votes]: It's not true.  Please accept this humble paint counter-example.

On the left, we see that the graph is 3-colorable.  On the right, we take the maximum cardinality independent set and make it all blue, then get stuck with the remaining two colors.