Independent set on colour critical graphs

Question

I want to prove the following property on graphs:

For every independent set $S$ of a colour critical graph $G$, the equality $\chi(G-S) = \chi(G)-1$ holds.

I cannot even intuitively understand why this property is true. I mean, $S$ can have a lots of vertices from $G$, so I don't see why removing any $S$ with the single and only property that the nodes in it are not adjacent between themselves decrements the chromatic colour only by $1$ and not even more.

I would really appreciate any ideas and intuitive explanation of why this property holds given the hypothesis on the problem. Thanks in advance

Answer 1

You can actually prove a stronger result: If $S$ is an independent set of a graph $G$, then $\chi(G-S) \geq \chi(G)-1$.

As an additional hint, suppose that $\chi(G-S) < \chi(G)-1$.

Could you use this to come up with a good coloring of $G$.