TITLE: Does a planar triangulation always contain a Hamiltonian path?
QUESTION [6 upvotes]: What about a Hamiltonian path in a triangulation of an n-gon? If not, how long is the longest path?

REPLY [8 votes]: There are two planar triangulations on 14 vertices without hamiltonian paths. This is the smallest size. For sure these are well-known.

Those answer the question for $n=3$. For $n=4$ the first examples appear at 12 vertices. Same for $n=5$. For $n=6$ one with 10 vertices. For $n=7$ one with 11 vertices.  And so forth.