Brooks' Theorem states that if $G$ is a connected graph then $\chi(G) \leqslant \Delta(G)$ unless $G$ is complete or is an odd cycle. Prove the theorem for 3-connected graphs $G$.

Let $G$ be a graph, and let $d_{1}+d_{2}=\Delta(G)-1$. By considering a partition $V_{1}, V_{2}$ of $V(G)$ that minimizes the quantity $d_{2} e\left(G\left[V_{1}\right]\right)+d_{1} e\left(G\left[V_{2}\right]\right)$, show that there is a partition with $\Delta\left(G\left[V_{i}\right]\right) \leqslant d_{i}, i=1,2$.

By taking $d_{1}=3$, show that if a graph $G$ contains no $K_{4}$ then $\chi(G) \leqslant \frac{3}{4} \Delta(G)+\frac{3}{2}$.