Regular Flip Equivalence of Surface Triangulations

Any two triangulations of a closed surface with the same number of vertices can be transformed into each other by a sequence of flips, provided the number of vertices exceeds a number N depending on the surface. Examples show that in general N is bigger than the minimal number of vertices of a triangulation. The existence of N was known, but no estimate. This paper provides an estimate for N that is linear in the Euler characteristic of the surface. 1. Results on flip equivalence Let F be a closed surface and let χ(F ) be its Euler characteristic. A singular triangulation of F is a graph T embedded in F such that each face of F \ T is bounded by an edge path of length three. We denote by v(T ), e(T ) and f(T ) the number of vertices, edges and faces of T . If T is without loops and multiple edges and has more than three faces, then T corresponds to a triangulation of F in the classical meaning of the word; in order to avoid confusions, we use for it the term regular triangulation. Let e be an edge of a singular triangulation T and suppose that there are two distinct faces δ1 and δ2 adjacent to e. The faces δ1 and δ2 form a (possibly degenerate) quadrilateral, containing e as a diagonal. A flip of T along e replaces e by the opposite diagonal of this quadrilateral, see Figure 1. The flip is called regular, if both T and the result of the flip are regular triangulations. Two singular (resp. regular) triangulations T1, T2 of a closed surface are called flip equivalent (resp. regularly flip equivalent), if they are related by a finite sequence of flips (resp. regular flips) and isotopy. The following result is well known, and there are many proofs for it. There are interesting applications to the automatic structure of mapping class groups, see [4] or [8]. Lemma 1. Any two singular triangulations T1 and T2 of a closed surface F with v(T1) = v(T2) are flip equivalent. One might ask whether any two regular triangulations of F with the same number of vertices are regularly flip equivalent. The answer is “Yes” in special cases: any two regular triangulations of the sphere [9], the torus [2], the projective plane or the Klein bottle [6] with the same number of vertices are regularly flip equivalent. But in general, the answer is “No”: it is known that there are 59 different triangulations of the closed oriented surface of genus six based on the complete graph with 12 vertices, see [1]. Such a triangulation does not admit any regular flip, thus the different triangulations are not regularly flip equivalent. This paper is devoted to the proof of the following theorem. A preliminary version of this paper has been appeared in [3]. Theorem 1. Let F be a closed surface and N(F ) = 9450 − 6020χ(F ). Any two regular triangulations T1 and T2 of F with v(T1) = v(T2) ≥ N(F ) are regularly flip equivalent.