Linear number of diagonal flips in triangulations on surfaces

A diagonal flip in a triangulation G on a surface is a transformation of G to replace a diagonal e in the quadrilateral region formed by two faces sharing e with another diagonal. If this operation breaks the simpleness of graphs, then we do not apply it. We shall prove that for any surface F 2 , there exists a natural number N ( F 2 ) such that if n ⩾ N ( F 2 ) , then any two n-vertex triangulations on F 2 can be transformed into each other by O ( n ) diagonal flips, up to homeomorphism.