ar X iv : m at h / 99 03 13 6 v 1 [ m at h . G T ] 2 3 M ar 1 99 9 An improvement of a result of Negami

Seiya Negami showed that any two triangulations of a closed surface with the same number of vertices can be transformed into each other by a sequence of regular flips, provided the number of vertices exceeds a number N depending on the surface. Negami's proof is inconstructive, he didn't give an estimate for N. It is shown in this paper that N is linearly bounded by the Euler characteristic of the surface. Let F be a compact surface without boundary and χ(F) its Euler characteristic. A singular triangulation T of F is a graph embedded in F such that any face is a triangle; if moreover T is without loops and multiple edges and if any two faces have at most one edge in common, then it is called regular triangulation. A contraction of a regular triangulation T along an edge e shrinks e to a vertex and eliminates the two faces adjacent to e, see Figure 1. The edge e is called contractible if the result of the contraction is still e Figure 1: Contraction along an edge a regular triangulation. A regular triangulation T is called irreducible triangulation if it does not contain contractible edges. Let e be an edge of a singular triangulation T and suppose there are two distinct faces δ 1 and δ 2 adjacent to e; the faces δ 1 and δ 2 form a quadrilateral, containing e as a diagonal. A flip of T along e replaces e by the opposite diagonal of this quadrilateral, see Figure 2. The flip is called a regular flip, if both T and the result of the flip are regular triangulations. Let v(T), e(T) and f (T) be the number of vertices, edges and faces, respectively, of a singular triangulation T. Proposition 1. Any two singular triangulations T 1 and T 2 of a closed surface F with v(T 1) = v(T 2) can be transformed into each other by a finite sequence of singular flips. There are many proofs for Proposition 1. I recommend Mosher's proof, since his techniques have interesting applications to the automatic structure of mapping class groups, see [Mos95] or [RW]. An open question is whether two regular triangulations with the same number of vertices can be transformed into each other by regular flips. The answer is " yes " in special 1