# Connection between chromatic number and independence number of a graph

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.]