Given a bicubic planar graph consisting of faces with degree $4$ and $6$, so called Barnette graphs.
We can show that there are exactly six squares.
Kundor and I found six types of arrangements of the six squares:
Is there anything known if any of these arrangements can be proven to be Hamiltonian?
For example, “3. two triples arranged like a triangle” gives much less structural degrees of freedom compared to “4. six isolated squares”…