Simultaneously Flippable Edges in Triangulations

Given a straight-line triangulation T , an edge e in T is flippable if e is adjacent to two triangles that form a convex quadrilateral. A set of edges E in T is simultaneously flippable if each edge is flippable and no two edges are adjacent to a common triangle. Intuitively, an edge is flippable if it may be replaced with the other diagonal of its quadrilateral without creating edge-edge intersections, and a set of edges is simultaneously flippable if they may be all be flipped without interferring with each other. We show that every straight-line triangulation on n vertices contains at least (n− 4)/5 simultaneously flippable edges. This bound is the best possible, and resolves an open problem by Galtier et al. .