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 ( n 2 ) . We also show that for general point sets, flip graphs of triangulations with degree ⩽k can be disconnected for any k.