So I've got an assignment to proof that the linegraph of a normal Hamiltonian graph is also Hamiltonian (but not Eulerian most of the time). I've found that this is a direct corollary of a theorem of Harary and Nash-Williams regarding dominating trails.
However, we've never discussed this theorem or dominating trails in class, so I doubt this is the answer they are looking for, so is there an alternative proof? I can't think, however, of any other ways to solve this question.
If $G$ is hamiltonian then we can arrange the vertices of the graph into a cycle. If $G$ has any other edges then they become chords of this cycle. If we want a hamiltonian cycle in $L(G)$, we need to be able to cycle through the edges of $G$ in such a way that consecutive edges in our sequence are incident in the graph. We can do that by moving along the edges of the cycle in $G$, and picking up any chordal edges along the way. See the example below, where the numbers on the edges indicate where they are in our sequence.
This paper by Chartrand brings up the concept of sequential graphs in determining when a line graph is hamiltonian, which is essentially the technique I presented here.