(i) State Brooks' Theorem, and prove it in the case of a 3 -connected graph.

(ii) Let $G$ be a bipartite graph, with vertex classes $X$ and $Y$, each of order $n$. If $G$ contains no cycle of length 4 show that

$$e(G) \leqslant \frac{n}{2}(1+\sqrt{4 n-3})$$

For which integers $n \leq 12$ are there examples where equality holds?