发表机构
TU Graz; TU Wien; TU Braunschweig; Utrecht University; Stenden University of Applied Sciences; FernUniversität in Hagen; Tulane University（格拉茨工业大学; 维也纳工业大学; 布伦瑞克工业大学; 乌得勒支大学; 斯滕登应用科学大学; 哈根远程大学; 杜兰大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究双色单位正方形瓷砖的完美矩形平铺问题，刻画六种瓷砖类型子集类别的可平铺性与判定复杂性。
AI 中文摘要
我们研究一个有限平铺问题，其中瓷砖是单位正方形，其四条边用两种颜色之一着色。我们询问给定的矩形是否允许完美矩形平铺：矩形的每个单元格被一个瓷砖占据，相邻边的颜色匹配，并且恰好使用 $n_i$ 个类型为 $i$ 的瓷砖，其中允许旋转瓷砖。我们的问题与经典的王氏平铺、更一般的有限瓷砖放置问题以及边缘放置谜题相关。但在我们的问题中，瓷砖类型的多重性是输入的一部分，且瓷砖字母表是固定且极小的；因此问题的复杂性源于矩形尺寸与规定的瓷砖多重性之间的相互作用。我们对完美矩形平铺问题进行了全面的研究。为此，我们考虑双色边的六种可能瓷砖类型的所有子集类别，并针对每个类别刻画多重性是否总是允许完美矩形平铺，或者其存在性是否可以高效判定。
英文摘要
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.
CommentsFull version of a paper with the same title to appear at ISAAC 2026
