AI 中文总结
本文研究公平着色问题的参数化复杂性，证明其对森林和模块宽度2图的组数为W[1]-困难，颜色数为2时对邻域多样性为FPT，一般情况对邻域多样性和组数为FPT，还证明一元向量装箱问题维度相关的W[1]-困难性。
AI 中文摘要
给定图$G=(V,E)$，图的（正常）$k$-着色是用$k$种颜色对顶点进行着色，使得任意两个相邻顶点颜色不同。假设顶点集$V$被划分为若干组，若每个颜色类中任意两组的顶点数之差不超过给定阈值，则该正常着色称为公平着色。本文研究公平着色问题关于输入图结构参数的参数化复杂性。具体而言，我们证明即使颜色数等于2，该问题对于森林和模块宽度为2的图，关于组数是W[1]-困难的；而在正面结果方面，我们证明当颜色数等于2时，该问题关于输入图的邻域多样性是固定参数可处理（FPT）的，且一般情况下，该问题关于邻域多样性和组数是FPT的。作为副产品，我们证明一元向量装箱问题关于维度是W[1]-困难的。
英文摘要
Given a graph $G=(V,E)$, a (proper) $k$-coloring for $G$ is a vertex coloring with $k$ colors such that every two adjacent vertices receive different colors. Suppose that the vertex set $V$ is partitioned into some groups, a proper coloring is called fair if for every color class, the difference between the number of vertices in any two groups does not exceed a given threshold. In this paper, we investigate the parameterized complexity of the fair coloring problem with respect to the structural parameters of the input graph. In particular, we prove that the problem is W[1]-hard with respect to the number of groups for forests and also graphs of modular-width two, even when the number of colors is equal to two. On the positive side, we prove that when the number of colors is equal to two, then the problem is FPT with respect to neighborhood diversity of the input graph. Moreover, in general, the problem is FPT with respect to neighborhood diversity and the number of groups. As a by-product, we prove that unary vector bin packing problem is W[1]-hard with respect to the dimension.
