Cubicity of Interval Graphs and the Claw Number

Abstract

Abstract Let G ( V , E ) be a simple, undirected graph. A b-dimensional box is a Cartesian product I 1 × I 2 × ⋯ × I b , where each I i is a closed interval on the real line. When each interval has unit length we have a b-dimensional cube. The cubicity (respectively boxicity) of G, cub ( G ) ( box ( G ) ) is the minimum positive integer b such that the vertices in G can be mapped to axis parallel b-dimensional cubes (boxes) in such a way that two vertices are adjacent in G if and only if their assigned cubes (boxes) intersect. Suppose S ( m ) denotes a star graph on m + 1 nodes. We define claw number ψ ( G ) to be the largest positive integer m such that S ( m ) is an induced subgraph of G. In this paper we show that for an interval graph G ⌈ log 2 ψ ( G ) ⌉ ⩽ cub ( G ) ⩽ ⌈ log 2 ψ ( G ) ⌉ + 2 . We also show that cub ( G ) ⩽ ⌈ log 2 α ⌉ , where α is the independence number of G. From this we have, for any graph G, cub ( G ) ⩽ box ( G ) ⌈ log 2 α ⌉ .