Regular flip equivalence of surface triangulations (Q1868890)

The author proves that any two triangulations of a closed surface~\(F\) with the same number of vertices can be transformed into each other by a sequence of diagonal flips, provided that the triangulations have at least \(N(F)=9450-6020\chi(F)\) vertices. This is the first explicit bound on the minimal necessary number \(\nu(F)\) of vertices, although the author conjectures it to be far from optimal. (Note that \(N(F)<0\) if and only if \(F\)~is a sphere, in which case the result was known classically.) In general, it is known (see \textit{A. Altshuler, J. Bokowski} and \textit{P. Schuchert} [J. Comb. Theory 75, 148-162 (1996; Zbl 0855.52002)]) that \(\nu(F)\)~is greater than the minimal number of vertices of a triangulation of~\(F\). Three main ingredients of the proof are: using the second barycentric subdivision; a result of \textit{A. Nakamoto} and \textit{K. Ota} [J. Graph Theory 20, 227-233 (1995; Zbl 0834.05021)] that bounds by \(270-171\chi(F)\) the number of vertices of any triangulation of~\(F\) that is irreducible with respect to contracting edges; and the following lemma of \textit{S. Negami} and \textit{S. Watanabe} [Tsukuba J. Math. 14, 155-166 (1990; Zbl 0719.05030)]: If the proper triangulation~\(T_2\) is obtained from the proper triangulation~\(T_1\) by contracting edges, then one can pass from~\(T_2\) to~\(T_1\) by subdividing \(f_0(T_1)-f_0(T_2)\) triangles of~\(T_2\) and executing proper diagonal flips. (Here \(f_0(T)\) denotes the number of vertices of~\(T\), and \textit{proper} triangulations consist of more than three faces, and their underlying graph has no loops or multiple edges.).