On the Boxicity and Cubicity of Hypercubes

Abstract

For a graph $G$, its \emph{cubicity} $cub(G)$ is the minimum dimension $k$ such that $G$ is representable as the intersection graph of (axis--parallel) cubes in $k$--dimensional space. Chandran, Mannino and Oriolo showed that for a $d$--dimensional hypercube $H_d$, $\frac{d-1}{\log d}\le cub(H_d)\le 2d$. In this paper, we show that $cub(H_d)=\Theta (\frac{d}{\log d})$.The parameter \emph{boxicity} generalizes cubicity: the boxicity $box(G)$ of a graph $G$ is defined as the minimum dimension $k$ such that $G$ is representable as the intersection graph of axis parallel boxes in $k$ dimensional space. Since $box(G)\le cub(G)$ for any graph $G$, our result implies that $box(H_d)=O(\frac{d}{\log d})$. The problem of determining a non-trivial lower bound for $box(H_d)$ is left open.