发表机构
Carleton University; University of Windsor; Simon Fraser University（卡尔顿大学; 温莎大学; 西蒙菲莎大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文改进三角剖分多色2-着色的差异上界至(3n-16)/7，并给出线性时间算法，结合适当四着色结果确认差异至多n/3。
AI 中文摘要
三角剖分的多色2-着色是对顶点进行2-着色，使得没有面是单色的。着色的差异是颜色类大小之间的最大差。Asayama和Matsumoto（Graphs and Combinatorics，2022）证明了每个三角剖分都允许差异至多$\ frac{5n-16}{9}$的多色2-着色，并且存在一类三角剖分，其中每个多色2-着色的差异至少为$\ frac{n}{3} - 2$，其中$n$是顶点数。我们改进了上界，证明每个三角剖分都允许差异至多$\ frac{3n-16}{7}$的多色2-着色，并且这样的2-着色可以在二次时间内计算。我们还证明了对于具有大小为$M$的匹配的三角剖分，差异至多为$n-\ frac{4M}{3}$。这例如意味着Delaunay三角剖分允许差异至多$\ frac{n}{3}$。我们提供了一个线性时间算法来计算差异至多$\ frac{5n-24}{7}$的2-着色。我们的一个结果表明，任何具有最大颜色类大小为$\ rac{n}{2}$的适当四着色都将意味着差异至多$\ rac{n}{3}$的2-着色。这种适当着色的存在最近由Kawarabayashi，Yoneda和Yoneda（arXiv 2026）确认。因此，这两个结果一起确认了三角剖分的差异至多$\ rac{n}{3}$。
英文摘要
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.
Comments14 pages, 3 figures, SWAT 2026
