Intersection Graph Of Axis-parallel Boxes Research Articles

Representing a Cubic Graph as the Intersection Graph of Axis-Parallel Boxes in Three Dimensions

We show that every graph of maximum degree 3 can be represented as the intersection graph of axis parallel boxes in three dimensions, that is, every vertex can be mapped to an axis parallel box such that two boxes intersect if and only if their corresponding vertices are adjacent. In fact, we construct a representation in which any two intersecting boxes touch just at their boundaries.

The cubicity of hypercube graphs

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.

On the cubicity of certain graphs

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.