# Smallest Planar Cubic Graph with Non Hamiltonian Edge

I'm looking for the smallest simple planar cubic hamiltonian graph without triangles and with at least one edge that never lies on a hamiltonian cycle.

I've got one with triangles $\phantom{somespaceneed}$ or one non planar example

It's obvious that the middle edge never lies on a Hamilton cycle, since If once you start down the [middle] dark path, forever will it dominate your destiny and you will never return to the origin without crossing one vertex twice. I'm not sure, if this approach (patching 2 blobs left and right of the non-HC edge) leads to the smallest solution.

May the Force be with you...

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