I know how to work out the chromatic polynomial of a graph and I can work out what $k$ would be by looking at the graph. Maybe I'm just being silly, but if you were given just the chromatic polynomial say: $$ k(k-1)^2 (k-2) $$ can you just look at that and tell that the chromatic number of $G$ is $3$?
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If $p(k)$ is the chromatic polynomial of $G$, then the chromatic number of $G$ is the smallest non-negative integer $k_0$ such that $p(k_0)>0$.
In this case, you can clearly see that $p(0)=p(1)=p(2)=0$, while $p(3)\neq 0$, so the chromatic number must be $3$.