Drawing a graph with chromatic polynomial

Question

I have chromatic polynomial,

$$P(k) = k^2 (k-1)^2 (k-2).$$

How can I draw a graph with given equation?

Answer 1

Let the vertices be $a,b,c,d,e$, and let the edges be $ab,bc,bd,cd$. Then there are $k$ ways to color vertex $a$, $k-1$ for $b$, $k-1$ for $c$, $k-2$ for $d$, and $k$ for $e$, so you get $k^2(k-1)^2(k-2)$.