I'm stuck with exercise 18.1.5 of Bondy & Murty's Graph Theory book which asks for an example of a 3-connected planar bipartite graph on fourteen vertices that is not traceable (that is, which has no Hamiltonian path).
I'm aware of Barnette's conjecture which states that every cubic 3-connected planar bipartite graph is Hamiltonian and I know that such a small counterexample on fourteen vertices does not exist, so if I'm right the 3-connected bipartite planar graph that I'm looking for has minimum degree at least three and it must have at least a vertex of degree greater than 3 but all the graphs that I have drawn fail to fulfill one of the conditions (planarity, 3-connectedness or being bipartite).
Even if such an example was found, I would still need to prove that it has no Hamiltonian path which doesn't seem easy either. For the examples that I was trying to build, I was considering all the possible paths going through a vertex to show that it is not traceable, but that's a very long task.
Any hints or ideas would be much appreciated.