Flip graphs of bounded-degree triangulations

We study flip graphs of triangulations whose maximum vertex degree is bounded by a constant k. Specifically, 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(n2). We also show that for general point sets, flip graphs of triangulations with degree k can be disconnected for any k