The cubicity of hypercube graphs

For a graph image, its cubicity image is the minimum dimension image such that image is representable as the intersection graph of (axis-parallel) cubes in image-dimensional space. (A image-dimensional cube is a Cartesian product image, where image is a closed interval of the form image on the real line.) Chandran et al. [L.S. Chandran, C. Mannino, G. Oriolo, On the cubicity of certain graphs, Information Processing Letters 94 (2005) 113–118] showed that for a image-dimensional hypercube image, image. In this paper, we use the probabilistic method to show that image. The parameter boxicity generalizes cubicity: the boxicity image of a graph image is defined as the minimum dimension image such that image is representable as the intersection graph of axis-parallel boxes in image-dimensional space. Since image for any graph image, our result implies that image. The problem of determining a non-trivial lower bound for image is left open.