An upper bound for Cubicity in terms of Boxicity

An axis-parallel b-dimensional box is a Cartesian product R1 ×R2 × . . .× Rb where each Ri (for 1 ≤ i ≤ b) is a closed interval of the form [ai, bi] on the real line. The boxicity of any graph G, box(G) is the minimum positive integer b such that G can be represented as the intersection graph of axis parallel b-dimensional boxes. A b-dimensional cube is a Cartesian product R1 × R2 × . . . × Rb, where each Ri (for 1 ≤ i ≤ b) is a closed interval of the form [ai,ai+1] on the real line. When the boxes are restricted to be axis-parallel cubes in b-dimension, the minimum dimension b required to represent the graph is called the cubicity of the graph (denoted by cub(G)). In this paper we prove that cub(G) ≤ ⌈logn⌉box(G) where n is the number of vertices in the graph. We also show that this upper bound is tight.