A polychromatic 2-coloring of a triangulation is a 2-coloring of the vertices such that no face is monochromatic. The discrepancy of a coloring is the maximum difference between the sizes of the color classes. Asayama and Matsumoto (Graphs and Combinatorics, 2022) proved that every triangulation admits a polychromatic 2-coloring with discrepancy at most 5n-169, and that there exists a class of triangulations for which every polychromatic 2-coloring has discrepancy at least n3 - 2, where n is the number of vertices. We improve the upper bound, showing that every triangulation admits a polychromatic 2-coloring with discrepancy at most 3n-167 and such a 2-coloring can be computed in quadratic time. We also show a discrepancy of at most n-4M3 for triangulations with a matching of size M. This implies, for example, that Delaunay triangulations admit a discrepancy of at most n3. We provide a linear-time algorithm to compute a 2-coloring whose discrepancy is at most 5n-247. One of our results shows that any proper four coloring with the largest color class of size n/2 would imply a 2-coloring with discrepancy at most n/3. The existence of such a proper coloring has been recently confirmed by Kawarabayashi, Yoneda, and Yoneda (arXiv 2026). Therefore the two results together confirm the discrepancy of at most n/3 for triangulations.
