发表机构
Indian Institute of Technology Jodhpur; Indian Institute of Science（印度技术学院乔德普尔分校; 印度科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究多数着色博弈的色数，证明无环图（最大度≤4）的多数博弈色数≤3，路径、星图、完全图≤2，并证明预着色扩展问题及其博弈版本分别为NP完全和PSPACE完全。
AI 中文摘要
图$G$的多数着色（也称为不友好划分）是$G$的一种顶点着色，使得每个顶点的邻居中，与其自身颜色相同的邻居数不超过其邻居总数的一半。$G$的多数着色所需的最少颜色数称为多数色数$\mu(G)$。多数着色博弈由Bosek--Grytczuk--Jakóbczak（2019）引入，是一种两人Maker--Breaker型博弈，其中玩家交替对顶点着色，同时保持每个顶点的多数条件。先手在$G$上具有必胜策略所需的最少颜色数称为多数博弈色数$\mu_g(G)$。与静态情形相反，Bosek等人证明$\mu_g(G)$在一般情况下无界，而$\mu_g(G) \le \mathrm{col}_g(G)$，其中$\mathrm{col}_g(G)$是$G$的博弈着色数。已知对于任何无环图$G$，$\mathrm{col}_g(G) \le 4$，因此$\mu_g(G) \le 4$。我们通过证明对于任何最大度至多为4的无环图$G$，$\mu_g(G) \le 3$，改进了这一界。我们还证明，若$G$是路径、星图或完全图，则$\mu_g(G) \le 2$，改进了Bosek等人的结果。我们还开始了对多数着色博弈计算复杂性的研究。我们证明，在$G$上使用$\chi(G)$种颜色的调色板的多数着色的预着色扩展问题是NP完全的，其博弈版本是PSPACE完全的。此外，即使调色板只有2种颜色，该问题仍是NP完全的，其博弈版本仍是PSPACE完全的。
英文摘要
A majority coloring of a graph $G$ is a vertex coloring of $G$ in which no vertex has more than half of its neighbors colored with its own color. The least number of colors required for a majority coloring of $G$ is the majority chromatic number $μ(G)$. The majority coloring game, introduced by Bosek et al. (2019), is a two-player Maker-Breaker-type game where the players alternately color vertices while maintaining the majority condition at each vertex; the least number of colors required for the first player to have a winning strategy on $G$ is the majority game chromatic number $μ_g(G)$. In contrast with the static case, Bosek et al. show that $μ_g(G)$ is unbounded in general, while $μ_g(G)\le col_g(G)$, where $col_g(G)$ is the game coloring number of $G$. It is known (cf. Faigle et al. (1993)) that, for any acyclic graph $G$, $col_g(G) \leq 4$, so $μ_g(G) \leq 4$ as well. We improve this bound by showing that $μ_g(G) \leq 3$ for any acyclic graph $G$ of maximum degree at most $4$. We also show that $μ_g(G) = 2$ if $G$ is a nonempty path, star, or complete graph, or a disjoint union of such graphs. These improve the results of Bosek et al. We also initiate the study of the computational complexity of the pre-coloring extension problem for the static and game versions of majority coloring. We show that the Majority $k$-precoloring Extension problem is NP-complete for each $k \geq 2$. When $k = 2$, it remains NP-complete on the class of bipartite graphs where maximum degrees of each part are $5$ and $6$, and when $k = 3$ it remains NP-complete on the class of $12$-regular planar graphs. We show that the game version is PSPACE-complete, even when the number of colors $k=χ(G)$, and even when $k=2$ and $G$ is a bipartite graph in which one part has maximum degree equal to $9$. The results on the game versions also hold for the variation where Bob starts the game.
Comments22 pages, 9 figures, accepted at FSTTCS 2026
