Sub-exponential algorithm for Hamiltonian cycle problem on cubic planar graphs?
Actually, you don't need to exploit the "Catalan structure" in planar graphs to get a $2^{O(\sqrt{n})}$ time bound as Saeed suggests. Cygan et al. show you how to detect a Hamiltonian Cycle in $\operatorname{poly}(n)4^{\operatorname{tw}(G)}$ time even in general graphs, and in the final version of the paper they also describe how this can be improved to $\operatorname{poly}(n)3^{\operatorname{tw}(G)}$ time in cubic graphs. 

The above algorithm is randomized. If you insist on a deterministic one, this is also possible with a slightly larger base constant, see Bodlaender et al.