Solution
= 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.