NP completeness of Hamiltonian cycle for the family of *dual graphs* to plane, cubic, triply connected graphs? It is well known that the Hamiltonian cycle problem is NP-complete on the family of planar, cubic and triply connected graphs: https://epubs.siam.org/doi/abs/10.1137/0205049 

For a problem I'm thinking about, it would be insightful to know if the Hamiltonian cycle problem was also NP-complete on the family of plane graphs whose dual is cubic and triply connected. 

I would be interested in any result of the following form:

Let $D_d = \{ G \text{ a planar graph} : G^* \text{ has max degree bounded by } d \}$. Then for some $d$, Hamiltonian cycle is NP complete on $D_d$.
The following paper shows that the Hamiltonian cycle problem is NP-complete in maximal planar graphs:


  A. Wigderson
  The Complexity of the Hamiltonian Circuit Problem for Maximal Planar Graphs
  Technical Report #298, Department of EECS, Princeton University, February 1982.
  https://www.math.ias.edu/avi/node/820


In a maximal planar graph, every face is a triangle; hence the dual of a  maximal planar graph is cubic. It is easy to see that the dual of a maximal planar graph is triply connected. 

Therefore your problem is NP-complete.