Diagonal Flips in Triangulations on Closed Surfaces with Minimum Degree at Least 4

It will be shown that any two triangulations on a closed surface, except the sphere, with minimum degree at least 4 can be transformed into each other by a finite sequence of diagonal flips through those triangulations if they have a sufficiently large and same number of vertices. The same fact holds for the sphere if they are not equivalent to a double wheelCn+K2.