Perfect Rectangular Tilings with Two Colors

2026-09-22

AI Summary

研究了使用两种颜色边缘的单位正方形瓷砖对矩形进行完美平铺的问题，考虑了瓷砖类型数量和矩形尺寸间的相互作用，并对所有可能的瓷砖子集进行了全面分析。

Abstract

We study a finite tiling problem, where tiles are unit squares whose four edges are colored with one of two colors. We ask whether a given rectangle admits a perfect rectangular tiling: every cell of the rectangle is occupied by one tile, neighboring edge colors match, and exactly $n_i$ tiles of type $i$ are used, where rotations of the tiles are allowed. Our problem is related to classical Wang tilings, more general finite tile-placement problems, and edge placement puzzles. But in our problem, the multiplicities of the tile types are part of the input and the tile alphabet is fixed and extremely small; thus the complexity of the problem arises from the interaction between the rectangle dimensions and the prescribed tile multiplicities. We provide a comprehensive study of the perfect rectangular tiling problem. For this we consider all classes of subsets of the six possible tile types for two-colored edges, and we characterize for each class whether multiplicities either always allow a perfect rectangular tiling or whether their existence can be decided efficiently.

Problem

• perfect rectangular tiling
• two colors
• tile multiplicities

Innovation

• perfect rectangular tiling
• two-color tiles
• tile multiplicities
• efficient decidability