TITLE: Hamiltonian circuit
QUESTION [9 upvotes]: Let us consider a disk with one labelled point on the boundary and $n$ labelled points in the interior. 
Let T be a triangulation of the whole disk with vertices on the labelled points such that T contains no self-loops except the boundary of the disk, no multiple edges between two points in the interior of T, arcs of T can be curved. 
1) Does T have an Hamiltonian circuit ?
2) If R is another triangulation of the same time, how many flips (as a function of  $ n $ should one perform at most in order to get from R to T ?

REPLY [2 votes]: Not necessarily.
See the famous paper of Sleator/Tarjan/Thurston (Journal of the AMS, vol 1, issue 1, I believe).