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 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 $$G$$ be a graph. Then the cubicity of $$G$$, denoted by $$\operatorname{cub} (G)$$, is the smallest integer $$n$$ such that $$G$$ can be realized as an intersection graph of axis-parallel unit cubes in $$n$$-dimensional Euclidean space.

The cubicity of a graph is closely related to the boxicity of a graph, denoted $$\operatorname{box} (G)$$. The definition of boxicity is essentially the same as cubicity, except in terms of using axis-parallel rectangles instead of 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. In the other direction, it can be shown that for any graph $$G$$ on $$m$$ vertices, the inequality $$\operatorname{cub} (G) \leq \lceil \log_2 n \rceil \operatorname{box} (G)$$, where $$\lceil x \rceil$$ is the ceiling function, i.e., the smallest integer greater than or equal to $$x$$.