An upper bound for Cubicity in terms of Boxicity

Abstract: An axis-parallel -dimensional box is a Cartesian product where each (for ) is a closed interval of the form on the real line. The boxicity of any graph , is the minimum positive integer such that G can be represented as the intersection graph of axis-parallel -dimensional boxes. A -dimensional cube is a Cartesian product , where each (for ) is a closed interval of the form on the real line. When the boxes are restricted to be axis-parallel cubes in -dimension, the minimum dimension required to represent the graph is called the cubicity of the graph (denoted by ). In this paper we prove that , where 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 . [2.] Outer planar graphs have cubicity at most . [3.] Any graph of treewidth has cubicity at most . Thus, chordal graphs have cubicity at most and circular arc graphs have cubicity at most , where is the clique number. The above upper bounds are tight, but for small constant factors. [Copyright &y& Elsevier]