(a) Let $G$ be a graph. What is a Hamilton cycle in $G$ ? What does it mean to say that $G$ is Hamiltonian?

(b) Let $G$ be a graph of order $n \geqslant 3$ satisfying $\delta(G) \geqslant \frac{n}{2}$. Show that $G$ is Hamiltonian. For each $n \geqslant 3$, exhibit a non-Hamiltonian graph $G_{n}$ of order $n$ with $\delta\left(G_{n}\right)=\left\lceil\frac{n}{2}\right\rceil-1$.

(c) Let $H$ be a bipartite graph with $n \geqslant 2$ vertices in each class satisfying $\delta(H)>\frac{n}{2}$. Show that $H$ is Hamiltonian. For each $n \geqslant 2$, exhibit a non-Hamiltonian bipartite graph $H_{n}$ with $n$ vertices in each class and $\delta\left(H_{n}\right)=\left\lfloor\frac{n}{2}\right\rfloor$.