Fault-Free Tileability of Rectangles, Cylinders, Tori, and M\"obius Strips with Dominoes

TL;DR

This paper investigates fault-free domino tilings on various surfaces including rectangles, cylinders, tori, and M"obius strips, providing comprehensive results on their tileability properties.

Contribution

It extends the study of fault-free tilings from rectangles to complex surfaces like cylinders, tori, and M"obius strips, offering complete characterizations.

Findings

Complete results for fault-free tilings on cylinders.

Complete results for fault-free tilings on tori.

Complete results for fault-free tilings on M"obius strips.

Abstract

Tilings are around us everywhere, and our curiosity draws us to study their properties. A tiling is a way of arranging pieces on a board, such that there is no space left uncovered, nor any space covered by more than one tile. In particular, we study fault-free tilings of boards with dominoes. To be fault-free every line that intersects the tiling must also intersect the interior of at least one of the tiles. Fault-free rectangular boards have been well studied, however we look at boards that are cylinders, tori, and M\"obius strips. Using dominoes we study the various shapes of boards and see how the tileability changes. We have complete results for cylinders, tori, and M\"obius strips.