Cubicity
In graph theory, cubicity is a graph invariant defined to be the smallest dimension such that a graph can be realized as an intersection graph of axis-parallel unit cubes in Euclidean space. Cubicity was introduced by Fred S. Roberts in 1969, along with a related invariant called boxicity that considers the smallest dimension needed to represent a graph as an intersection graph of axis-parallel rectangles in Euclidean space.
Definition
Let be a graph. Then the cubicity of, denoted by, is the smallest integer such that can be realized as an intersection graph of axis-parallel unit cubes in -dimensional Euclidean space.The cubicity of a graph is closely related to the boxicity of a graph, denoted by. The definition of boxicity is essentially the same as cubicity, except in terms of using axis-parallel rectangles instead of axis-parallel unit cubes. Since a cube is a special case of a rectangle, the cubicity of a graph is always an upper bound for the boxicity of a graph, i.e.,. In the other direction, it can be shown that for any graph on vertices,, where is the ceiling function, i.e., the smallest integer greater than or equal to.