发表机构
Indian Institute of Technology Jodhpur; Indian Institute of Science; Amrita Vishwa Vidyapeetham（印度理工学院焦浦尔分校; 印度科学学院; 阿姆里塔大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究多数染色博弈的结构与计算复杂性，证明多种图类上 Alice 可用至多 3 色获胜，并精确计算环的强多数博弈色数，同时证明相关判定问题为 PSPACE-完全或 NP-完全。
AI 中文摘要
图 $G = (V,E)$ 的多数染色是对 $V(G)$ 的一种染色，使得对每个顶点 $v$，与 $v$ 同色的邻居数量至多为 $deg(v)/2$。强多数染色是对 $V(G)$ 的一种染色，使得对每个顶点 $v$，$N(v)$ 中每个单色子集的大小至多为 $deg(v)/2$。（强）多数染色博弈是一种两人 Maker-Breaker 型博弈，其中两位玩家 Alice 和 Bob 轮流对图 $G$ 的顶点进行染色，并保持（强）多数条件。使得 Alice 在此类博弈中拥有必胜策略的最少颜色数称为图 $G$ 的（强）多数博弈色数，记为 $\mu_g(G)$（强版本记为 $\mathrm{Maj}_g(G)$）。对于多数染色博弈，我们证明在以下情况下 $\mu_g(G) \le 3$：$G$ 是 $2$-毛毛虫图，$G$ 是所有叶子深度 $k \le 4$ 的有根树，以及 $G$ 是某个图的细分图。后者解决了 Bosek--Grytczuk--Jakóbczak 在 2019 年提出的一个问题，他们还询问是否对每棵树 $T$ 都有 $\mu_g(T) \le 3$。对于后一个问题，我们讨论了当 Alice 尝试在树上使用自然策略来赢得多数染色博弈时出现的各种困难。我们将其与树上的标记博弈和松弛染色博弈，以及局部有限无环图 $G$（其中 $\delta(G) > 1$）上的多数染色博弈进行了比较。对于强多数染色博弈，我们精确计算了每个环 $C_n$（$n \ge 3$）的 $\mathrm{Maj}_g(C_n)$。我们还启动了强多数染色博弈的计算复杂性的研究；具体来说，我们证明了强多数博弈色数问题的判定版本是 PSPACE-完全的。我们还证明了强多数 2-染色问题在欧拉图上是 NP-完全的。
英文摘要
A majority coloring of a graph $G = (V,E)$ is a coloring of $V(G)$ such that, for each vertex $v$, the number of neighbors of $v$ with the same color as $v$ is at most $deg(v)/2$. A strong majority coloring is a coloring of $V(G)$ such that, for each vertex $v$, every monochromatic subset of $N(v)$ has size at most $deg(v)/2$. The (strong) majority coloring game is a two-player Maker-Breaker-type game, in which two players Alice and Bob color the vertices of a graph $G$ alternately, maintaining the (strong) majority condition. The least number of colors such that Alice has a winning strategy in such a game is called the (strong) majority game chromatic number of the graph $G$, denoted $μ_g(G)$ (or $\mathrm{Maj}_g(G)$ for the strong version). For the majority coloring game, we prove that $μ_g(G) \le 3$ under the following cases: $G$ is a $2$-caterpillar, $G$ is a rooted tree with all leaves at depth $k \le 4$, and $G$ is a subdivision of some graph. The latter resolves a problem posed by Bosek--Grytczuk--Jakóbczak in 2019, who also asked whether $μ_g(T) \le 3$ for every tree $T$. For the latter question, we discuss various difficulties that arise when natural strategies are attempted by Alice to win the majority coloring game on trees. We include a comparison with the marking game and relaxed coloring game on trees, and with the majority coloring game on locally finite acyclic graphs $G$ with $δ(G) > 1$. For the strong majority coloring game, we compute $\mathrm{Maj}_g(C_n)$ exactly for each cycle $C_n$, $n \ge 3$. We also initiate the study of the computational complexity of the strong majority coloring game; specifically, we prove that the decision version of the Strong Majority Game Chromatic Number problem is PSPACE-complete. We also show that the Strong Majority 2-Coloring problem is NP-complete on Eulerian graphs.
Comments47 pages, 22 figures, 2 tables
