Cubicity, boxicity, and vertex cover

A k-dimensional box is the cartesian product R1 ×R2 × · · · ×Rk where each Ri is a closed interval on the real line. The boxicity of a graph G, denoted as box(G), is the minimum integer k such that G is the intersection graph of a collection of k-dimensional boxes. A unit cube in k-dimensional space or a k-cube is defined as the cartesian product R1 × R2 × · · · × Rk where each Ri is a closed interval on the real line of the form [ai, ai + 1]. The cubicity of G, denoted as cub(G), is the minimum k such that G is the intersection graph of a collection of k-cubes. In this paper we show that cub(G) ≤ t+ ⌈log (n− t)⌉− 1 and box(G) ≤ ⌊