Cubicity

A cubicity 2 graph realized as the intersection graph of unit cubes, i.e. squares, in the plane.

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 $\mathrm{cub}(G)$, 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 $\mathrm{box}(G)$. 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 $\mathrm{cub}(G)\le \lceil {\log }_{2}n\rceil \mathrm{box}(G)$, where $\lceil x\rceil$ is the ceiling function, i.e., the smallest integer greater than or equal to $x$.