$$N$$ N -Flips in 4-Connected Even Triangulations on the Sphere

Abstract

An even triangulation is a plane triangulation in which each vertex has even degree. It is well known that every even triangulation G has a unique 3-coloring, where the decomposition of the vertices of G by the 3-coloring is called the tripartition of G. Nakamoto et al. (Graph Theory 51:260-268, 2006) proved that every two even triangulations with the same tripartition can be transformed into each other by Nflips, where an N -flip is an operation transforming an even triangulation into an even triangulation. In this paper, we prove that every two 4-connected even triangulations G and G with the same tripartition can be transformed into each other by N -flips unless G or G is isomorphic to a generalized octahedron.