  We study flip graphs of triangulations whose maximum vertex degree is bounded
by a constant $k$. In particular, we consider triangulations of sets of $n$
points in convex position in the plane and prove that their flip graph is
connected if and only if $k > 6$; the diameter of the flip graph is $O(n^2)$.
We also show that, for general point sets, flip graphs of pointed
pseudo-triangulations can be disconnected for $k \leq 9$, and flip graphs of
triangulations can be disconnected for any $k$. Additionally, we consider a
relaxed version of the original problem. We allow the violation of the degree
bound $k$ by a small constant. Any two triangulations with maximum degree at
most $k$ of a convex point set are connected in the flip graph by a path of
length $O(n \log n)$, where every intermediate triangulation has maximum degree
at most $k+4$.
