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

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)
• Since line graphs tend to have a hamilton-cycle, I wonder if there is a classification of graphs with a line graph without a hamilton cycle. – Peter Oct 2 '14 at 10:01
• Have you checked this paper sciencedirect.com/science/article/pii/S0893965912001036 dealing with a conjecture of Thomassen, stating that every $4$-connected line graph is hamiltonian? Its not really what you're asking for but it may offer some insight. – Jernej Oct 2 '14 at 10:09
• I did not read the paper, but I noticed the conjecture. – Peter Oct 2 '14 at 10:11
• Is the line graph of a $2$-connected graph always $2$-connected ? – Peter Oct 2 '14 at 10:13
• Hm.. what is surely true is that if $G$ is $2$-edge connected then $L(G)$ is $2$-connected. This may give a clue for counterexamples. – Jernej Oct 2 '14 at 10:19

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