The cubicity of hypercube graphs

Abstract

For a graph G , its 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. (A k -dimensional cube is a Cartesian product R 1 × R 2 × ⋯ × R k , where R i is a closed interval of the form [ a i , a i + 1 ] 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 d -dimensional hypercube H d , d − 1 log d ≤ cub ( H d ) ≤ 2 d . In this paper, we use the probabilistic method to show that cub ( H d ) = Θ ( d log d ) . The parameter 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 ) ≤ cub ( G ) for any graph G , our result implies that box ( H d ) = O ( d log d ) . The problem of determining a non-trivial lower bound for box ( H d ) is left open.