On the Tileability of Polygons with Colored Dominoes

Abstract

We consider questions concerning the tileability of orthogonal polygons with colored dominoes. A colored domino is a rotatable $2 imes 1$ rectangle that is partitioned into two unit squares, which are called faces, each of which is assigned a color. In a colored domino tiling of an orthogonal polygon $P$, a set of dominoes completely covers $P$ such that no dominoes overlap and so that adjacent faces have the same color. We demonstrated that for simple layout polygons that can be tiled with colored dominoes, two colors are always sufficient. We also show that for tileable non-simple layout polygons, four colors are always sufficient and sometimes necessary. We describe an $O(n)$ time algorithm for computing a colored domino tiling of a simple orthogonal polygon, if such a tiling exists, where $n$ is the number of dominoes used in the tiling. We also show that deciding whether or not a non-simple orthogonal polygon can be tiled with colored dominoes is NP-complete.