3-Colored Triangulation of 2D Maps.

We describe an algorithm to triangulate a general map on an arbitrary surface in such way that the resulting triangulation 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 4e − 2b triangles when applied to a map with e edges, such that b of them are border edges (adjacent to only one face). Our algorithm yields a subdivision with at most 2e − b + 2(2 − χ) triangles, where χ is the Euler Characteristic of the surface; or at most 2e − b + 2(2 − χ) − n + (2(2 − χ) − b)/(D − 2) triangles if all n faces of the map have the same degree D. Experimental results show that the resulting triangulations have, on the average, significantly fewer triangles than these upper bounds.