Economic 3-Colored Subdivision of Triangulations

Abstract

We describe an algorithm to subdivide an arbitrary triangulation of a surface to produce a triangulation that is vertex-colorable with three colors. (Three-colorable triangulations can be efficiently represented and manipulated by the GEM data structure of Montagner and Stolfi.) The standard solution to this problem is the barycentric subdivision, which produces 6n triangles when applied to a triangulation with n faces. Our algorithm yields a subdivision with at most 2n − m + 4(2 − χ) triangles, where χ is the Euler characteristic of the surface and m is the number of border edges (adjacent to only one triangle). This bound is rarely reached in practice; in particular, if the triangulation is already three-colorable the algorithm does not split any triangles.