Let $G$ be a graph with $|G| \geqslant 3$. State and prove a necessary and sufficient condition for $G$ to be Eulerian (that is, for $G$ to have an Eulerian circuit).

Prove that if $\delta(G) \geqslant|G| / 2$ then $G$ is Hamiltonian (that is, $G$ has a Hamiltonian circuit).

The line graph $L(G)$ of $G$ has vertex set $V(L(G))=E(G)$ and edge set

$$E(L(G))=\{e f: e, f \in E(G), e \text { and } f \text { are incident }\} .$$

Show that $L(G)$ is Eulerian if $G$ is regular and connected.

Must $L(G)$ be Hamiltonian if $G$ is Eulerian? Must $G$ be Eulerian if $L(G)$ is Hamiltonian? Justify your answers.