(i) show that if G is a hamiltonian bipartite graph with partite sets $\displaystyle V_{1}$ and $\displaystyle V_{2}$ then $\displaystyle |V_{1}| = |V_{2}|$

(ii) let $\displaystyle G$ and $\displaystyle H$ be any two hamiltonian graphs, show that the graph $\displaystyle \overline{K_{2}} + (G \cup H)$ is hamiltonian