Prove that every graph $G$ on $n \geqslant 3$ vertices with minimal degree $\delta(G) \geqslant \frac{n}{2}$ is Hamiltonian. For each $n \geqslant 3$, give an example to show that this result does not remain true if we weaken the condition to $\delta(G) \geqslant \frac{n}{2}-1$ ( $n$ even) or $\delta(G) \geqslant \frac{n-1}{2}$ ( $n$ odd).

Now let $G$ be a connected graph (with at least 2 vertices) without a cutvertex. Does $G$ Hamiltonian imply $G$ Eulerian? Does $G$ Eulerian imply $G$ Hamiltonian? Justify your answers.