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. (Threecolorable 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).

more »

... 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 − n − 2b + 4(2 − χ) triangles if all n faces of the map have the same degree. Experimental results show that the resulting triangulations have, on the average, significantly fewer triangles than these upper bounds. * Institute of Computing, University of Campinas, 13081-970 Campinas, SP, Brazil. Financial support from FAPESP, process 2012/14698-7 3-Colored Triangulation 1. Let M ← M and T ← {}. Choose any face m of M and P rocess(m, T , M ).