Bounding the number of holes required for folding rectangular polyomoinoes into cubes

Abstract: We study the problem of whether rectangular polyominoes with holes are cube-foldable, that is, whether they can be folded into a cube, if creases are only allowed along grid lines. It is known that holes of sufficient size guarantee that this is the case. Smaller holes which by themselves do not make a rectangular polyomino cube-foldable can sometimes be combined to create cube-foldable polyominoes. We investigate minimal sets of holes which guarantee cube-foldability. We show that if all holes are of the same type, the these minimal sets have size at most 4, and if we allow different types of holes, then there is no upper bound on the size.