# hamiltonian graphs

(i) show that if G is a hamiltonian bipartite graph with partite sets $V_{1}$ and $V_{2}$ then $|V_{1}| = |V_{2}|$
(ii) let $G$ and $H$ be any two hamiltonian graphs, show that the graph $\overline{K_{2}} + (G \cup H)$ is hamiltonian