On the cubicity of certain graphs

Abstract

The boxicity of a graph G is the minimum dimension b such that G is representable as the intersection graph of axis-parallel boxes in the b-dimensional space. When the boxes are restricted to be axis-parallel b-dimensional cubes, the minimum dimension b required to represent G is called the cubicity of G. In this paper we show that cubicity ( H d ) ⩽ 2 d , where H d is the d-dimensional hypercube. (The d-dimensional hypercube is the graph on 2 d vertices which corresponds to the 2 d d-vectors whose components are either 0 or 1, two of the vertices being adjacent when they differ in just one coordinate.) We also show that cubicity ( H d ) ⩾ ( d − 1 ) / ( log d ) . We also show that (1) cubicity ( G ) ⩾ ( log α ) / ( log ( D + 1 ) ) , (2) cubicity ( G ) ⩾ ( log n − log ω ) / ( log D ) , where α , ω , D and n denote the stability number, the clique number, the diameter and the number of vertices of G. As consequences of these lower bounds we provide lower bounds for the cubicity of planar graphs, bipartite graphs, triangle-free graphs, etc., in terms of their diameter and the number of vertices.