2×2卡片对方形网格的无冗余覆盖：一个缺陷框架

中文摘要 AI 辅助

我们研究方形网格的一种无冗余覆盖，其中卡片可以重叠且每张卡片必须完全位于网格内部；网格的每个方格必须被覆盖，且每张卡片必须至少有一个仅由它自身覆盖的方格。我们为使用2×2卡片对方形网格进行无冗余覆盖开发了一个通用框架。我们首先引入一种将卡片划分为不相交集合的方法，并利用容斥原理及其他组合论证，推导出网格无冗余覆盖最大规模的一个宽松上界。在此宽松上界之后，我们开发了一个将网格方格划分为不相交类别的框架，并利用双重计数及其他方法推导出一个更紧的上界。随后，我们开发了一个统一的计数框架，并利用边界强制、局部重叠和缺陷论证得出一个严格上界。我们通过展示一个具有相同首项系数的下界，证明该上界是一个严格的渐近界。10×10网格仅作为动机和基准，而非新颖性声明；新颖性声明在于无冗余覆盖的通用框架。

英文摘要

We study an irredundant covering of an $m \times m$ square grid by $2 \times 2$ cards, where cards may overlap and every card must lie entirely inside the grid. Every square of the grid must be covered, and every card must contain at least one square that is covered only by itself. Let \(F(m)\) denote the maximum number of cards in such an irredundant cover. We develop a general framework for bounding the maximum possible size of such an irredundant cover based on square multiplicities, double counting, and local geometric restrictions arising around highly covered squares. These arguments yield a general upper bound whose leading term is $\frac{1}{2}m^2$, accompanied by a negative linear term. In the opposite direction, we construct a family of irredundant covers based on a density-$\frac{1}{2}$ covering pattern, referred to as the period-four staircase pattern, together with a boundary-repair construction. Thus, we obtain a lower bound with the same leading term and a linear-order error, proving that for every \(m\ge6\), \[ \frac{2m}{9} \le \frac12m^2-F(m) \le 2m-3, \] and consequently that the maximum density of an irredundant cover tends to $\frac{1}{2}$ as $m\to\infty$. The known $10\times 10$ case is used as motivation and as a finite benchmark for the general theory.

发表机构

• Dokuz Eylül University（德格埃布尔大学）

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