发表机构
The Omega Institute; ChronoAI Pte Ltd; Department of Fundamental Science and Technology National Higher School of Advanced Technologies; National University of Singapore（欧米伽研究所; ChronoAI私人有限公司; 国家高等先进技术学院基础科学与技术系; 新加坡国立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究图的子拟群染色数，建立结构界、关联2-独立数、分析计算复杂性，并确定网格图与超立方体的精确值，提出一般等式猜想。
AI 中文摘要
子拟群染色是一种部分顶点染色，其中每个被染色的顶点在其染色的闭邻域中至少有一半的顶点与其颜色相同。该概念由Hedetniemi、Hedetniemi、Laskar和Mulder在其关于拟群染色的开创性工作中作为开放方向提出。我们进一步发展了由Hedetniemi、Hedetniemi、Laskar和Mulder引入的子拟群染色数$\psq(G)$的研究。我们的重点在于结构界、计算复杂性、网格图和超立方体。我们建立了通用界，将$\psq$与$2$-独立数联系起来，讨论了计算复杂性，确定了几个经典族的确切值，并给出了网格条带的精确和计算机辅助结果。我们还研究了超立方体。特别地，对于$Q_n$，$2\le n\le6$的精确证书给出$\psq(Q_n)=\bii(Q_n)=2^{n-1}$，我们将一般等式表述为一个猜想。计算声明使用整数算术，并且可由稿件附带的验证器独立复现。
英文摘要
A sub-quorum coloring is a partial vertex coloring in which every colored vertex sees at least half of its colored closed neighborhood in its own color. Hedetniemi, Hedetniemi, Laskar and Mulder introduced its maximum number of colors, $\psq(G)$, as an open direction in their foundational work on quorum colorings. We establish general bounds, relate $\psq$ to $2$-independence, discuss computational complexity, and determine exact values for several classical families. For rectangular grids $G_{m,n}=P_m\square P_n$, we give a new profile proof of the known dissociation-number formula, equivalent to earlier exact $3$-path vertex-cover results. The proof supplies equality and rigidity information used to establish the same formula for the auxiliary parameter when the representative matching is restricted to one direction. We also obtain a five-sixths inequality for mixed-direction matchings on even-by-even rectangles. Exact transfer certificates establish the sub-quorum coloring formula for all fixed strip widths $2\le m\le11$. For hypercubes, we prove the dimension-free identity $\psq(Q_n)=\bii(Q_n)=2^{n-1}$ for every $n\ge2$. The upper bound for the sub-quorum coloring number follows from Huang's signed adjacency matrix through a restricted energy estimate and an injective linear map. The computer-assisted grid claims use integer arithmetic and are independently reproducible by the accompanying verifier.
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