Prove that every graph $G$ on $n \geqslant 3$ vertices with minimum 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 $\delta(G) \geqslant \frac{n}{2}-1$ (for $n$ even) or $\delta(G) \geqslant \frac{n-1}{2}$ (for $n$ odd).

For any graph $G$, let $G_{k}$ denote the graph formed by adding $k$ new vertices to $G$, all joined to each other and to all vertices of $G$. By considering $G_{1}$, show that if $G$ is a graph on $n \geqslant 3$ vertices with $\delta(G) \geqslant \frac{n-1}{2}$ then $G$ has a Hamilton path (a path passing through all the vertices of $G$ ).

For each positive integer $k$, exhibit a connected graph $G$ such that $G_{k}$ is not Hamiltonian. Is this still possible if we replace 'connected' with '2-connected'?