Past exam of the graph Theory and Probabilistic Combinatorics course of the University of Oxford / 2008 / A / 2 / e / Solution

The chromatic number is the smallest positive integer $q$ for which $p_G(q)>0$. Thus equality of the polynomials implies $\chi (G)=r$, as well as $|V(G)|=n$ and $e(G)=e(T_r(n))$. In particular $G$ contains no $K_{r+1}$, since such a clique requires $r+1$ colors. The equality case in Turán's theorem now gives $\boxed{G\cong T_r(n)}$. This uses more than equality of triangle counts: the polynomial also determines colorability.