Cubicity of interval graphs and the claw number

TL;DR

This paper establishes bounds on the cubicity of interval graphs in terms of the claw number and independence number, revealing that cubicity is closely related to these graph parameters and providing exact values in special cases.

Contribution

The paper derives tight bounds on the cubicity of interval graphs based on the claw number and independence number, and connects cubicity with boxicity.

Findings

Cubicity of interval graphs is between loor{ loor{ Cubicity equals loor{ when claw number equals independence number.

Cubicity is at most loor{ times boxicity for any graph.

Abstract

Let $G(V,E)$ be a simple, undirected graph where $V$ is the set of vertices and $E$ is the set of edges. A $b$-dimensional cube is a Cartesian product $I_1\times I_2\times...\times I_b$, where each $I_i$ is a closed interval of unit length on the real line. The \emph{cubicity} of $G$, denoted by $\cub(G)$ is the minimum positive integer $b$ such that the vertices in $G$ can be mapped to axis parallel $b$-dimensional cubes in such a way that two vertices are adjacent in $G$ if and only if their assigned cubes intersect. Suppose $S(m)$ denotes a star graph on $m+1$ nodes. We define \emph{claw number} $\psi(G)$ of the graph to be the largest positive integer $m$ such that $S(m)$ is an induced subgraph of $G$. It can be easily shown that the cubicity of any graph is at least $\ceil{\log_2\psi(G)}$. In this paper, we show that, for an interval graph $G$ $\ceil{\log_2\psi(G)}\le\cub(G)\le\ceil{\log_2\psi(G)}+2$. Till now we are unable to find any interval graph with $\cub(G)>\ceil{\log_2\psi(G)}$. We also show that, for an interval graph $G$, $\cub(G)\le\ceil{\log_2\alpha}$, where $\alpha$ is the independence number of $G$. Therefore, in the special case of $\psi(G)=\alpha$, $\cub(G)$ is exactly $\ceil{\log_2\alpha}$. The concept of cubicity can be generalized by considering boxes instead of cubes. A $b$-dimensional box is a Cartesian product $I_1\times I_2\times...\times I_b$, where each $I_i$ is a closed interval on the real line. The \emph{boxicity} of a graph, denoted $ box(G)$, is the minimum $k$ such that $G$ is the intersection graph of $k$-dimensional boxes. It is clear that $ box(G)\le\cub(G)$. From the above result, it follows that for any graph $G$, $\cub(G)\le box(G)\ceil{\log_2\alpha}$.