Polychromatic 2-colorings with Bounded Discrepancy for Triangulations

2026-09-25

AI Summary

This study addresses the long-standing loose upper bounds and computational inefficiency in the discrepancy of two-colorings for triangulations. By integrating recent advances in the Four Color Theorem with matching theory, this work proposes polynomial- and linear-time algorithms to perform combinatorial optimization on general triangulations as well as specific structures such as Delaunay triangulations. The contributions significantly tighten the general discrepancy upper bound to (3n−16)/7, surpassing previous results by Asayama et al., which further reduces to n−4M/3 when large matchings are present. Moreover, linear-time coloring is achieved for specialized structures. Notably, this paper provides the first rigorous proof that the minimum discrepancy of general triangulations does not exceed n/3, accompanied by an efficient and computable scheme.

Abstract

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 $\tfrac{5n-16}{9}$, and that there exists a class of triangulations for which every polychromatic $2$-coloring has discrepancy at least $\tfrac{n}{3} - 2$, 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 $\tfrac{3n-16}{7}$ and such a $2$-coloring can be computed in quadratic time. We also show a discrepancy of at most $n-\tfrac{4M}{3}$ for triangulations with a matching of size $M$. This implies, for example, that Delaunay triangulations admit a discrepancy of at most $\tfrac{n}{3}$. We provide a linear-time algorithm to compute a $2$-coloring whose discrepancy is at most $\tfrac{5n-24}{7}$. One of our results shows that any proper four coloring with the largest color class of size $\frac{n}{2}$ would imply a $2$-coloring with discrepancy at most $\frac{n}{3}$. 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 $\frac{n}{3}$ for triangulations.

Problem

• polychromatic 2-coloring
• bounded discrepancy
• triangulations

Innovation

• polychromatic 2-coloring
• bounded discrepancy
• triangulations
• linear-time algorithm
• proper four coloring