Let $G$ be a graph of order $n \geqslant 3$ satisfying $\delta(G) \geqslant \frac{n}{2}$. Show that $G$ is Hamiltonian.

Give an example of a planar graph $G$, with $\chi(G)=4$, that is Hamiltonian, and also an example of a planar graph $G$, with $\chi(G)=4$, that is not Hamiltonian.

Let $G$ be a planar graph with the property that the boundary of the unbounded face is a Hamilton cycle of $G$. Prove that $\chi(G) \leqslant 3$.