$2$-connected graphs with a line graph containing no hamilton cycle

Question

Let G be a simple undirected graph.

I found some examples of connected graphs G with line graphs containing no hamilton cycle, but none of them was $2$-connected.

• Are there $2$-connected graphs with a line graph containing no hamilton-cycle ?
• If yes, what is the smallest ? (Due to my search, it should have more than $7$ vertices)

Answer 1

Here is a counterexample on (incidentally) 8 vertices. I guess this can be generalized.

The idea is to only look for $2$-connected non-hamiltonian graphs since the line graph of a hamiltonian graph is always hamiltonian.

A natural question now is whether one can find a different kind of such graphs (not Theta graphs)