On the Cubicity of Interval Graphs

Abstract

A k-cube (or “a unit cube in k dimensions”) is defined as the Cartesian product $R_1 \times \cdots \times R_k$ where R _i (for 1 ≤ i ≤ k) is an interval of the form [a _i , a _i  + 1] on the real line. The k-cube representation of a graph G is a mapping of the vertices of G to k-cubes such that the k-cubes corresponding to two vertices in G have a non-empty intersection if and only if the vertices are adjacent. The cubicity of a graph G, denoted as cub(G), is defined as the minimum dimension k such that G has a k-cube representation. An interval graph is a graph that can be represented as the intersection of intervals on the real line - i.e., the vertices of an interval graph can be mapped to intervals on the real line such that two vertices are adjacent if and only if their corresponding intervals overlap. We show that for any interval graph G with maximum degree Δ, ${\rm cub} (G) \leq\lceil\log_2 \Delta\rceil +4$. This upper bound is shown to be tight up to an additive constant of 4 by demonstrating interval graphs for which cubicity is equal to $\lceil\log_2 \Delta\rceil$.