An upper bound for Cubicity in terms of Boxicity

Abstract

An axis-parallel b -dimensional box is a Cartesian product R 1 × R 2 × ⋯ × R b where each R i (for 1 ≤ i ≤ b ) is a closed interval of the form [ a i , b i ] 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 R 1 × R 2 × ⋯ × R b , where each R i (for 1 ≤ i ≤ b ) is a closed interval of the form [ a i , a i + 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 ) ≤ ⌈ log 2 n ⌉ box ( G ) , where n is the number of vertices in the graph. We also show that this upper bound is tight. Some immediate consequences of the above result are listed below: 1. Planar graphs have cubicity at most 3 ⌈ log 2 n ⌉ . 2. Outer planar graphs have cubicity at most 2 ⌈ log 2 n ⌉ . 3. Any graph of treewidth t w has cubicity at most ( t w + 2 ) ⌈ log 2 n ⌉ . Thus, chordal graphs have cubicity at most ( ω + 1 ) ⌈ log 2 n ⌉ and circular arc graphs have cubicity at most ( 2 ω + 1 ) ⌈ log 2 n ⌉ , where ω is the clique number. The above upper bounds are tight, but for small constant factors.