I'm working on an unassessed course problem,
A graph $H$ has chromatic polynomial $$P_H(k)=k(k-1)(k-2)^3(k-3)^2(k^2-4k+6).$$ What is the chromatic number of $H$? Prove that the chromatic index of $H$ is at least $5$.
I see that $$P_H(0)=P_H(1)=P_H(2)=P_H(3)=0<P_H(4) \hspace{1em} \therefore \hspace{1em} \chi_H=4.$$ The only approach I can think of to show $5\leq\chi_H'$ is to show that $5\leq\Delta(H)$ and apply Vizing's Theorem, but I don't know how to do that given only this information. Could someone give me a pointer?