Boxicity and Cubicity of Asteroidal Triple free graphs

TL;DR

This paper investigates the boxicity and cubicity of AT-free graphs, establishing bounds related to chromatic number and claw number, with special cases for graphs with girth at least 5.

Contribution

It provides new bounds on boxicity and cubicity of AT-free graphs based on chromatic and claw numbers, including tight bounds and special cases.

Findings

Boxicity of AT-free graphs is at most their chromatic number.

Cubicity is bounded by boxicity times a logarithmic function of claw number.

For AT-free graphs with girth ≥ 5, boxicity is at most 2.

Abstract

An axis parallel $d$-dimensional box is the Cartesian product $R_1 \times R_2 \times ... \times R_d$ where each $R_i$ is a closed interval on the real line. The {\it boxicity} of a graph $G$, denoted as $\boxi(G)$, is the minimum integer $d$ such that $G$ can be represented as the intersection graph of a collection of $d$-dimensional boxes. An axis parallel unit cube in $d$-dimensional space or a $d$-cube is defined as the Cartesian product $R_1 \times R_2 \times ... \times R_d$ where each $R_i$ is a closed interval on the real line of the form $[a_i,a_i + 1]$. The {\it cubicity} of $G$, denoted as $\cub(G)$, is the minimum integer $d$ such that $G$ can be represented as the intersection graph of a collection of $d$-cubes. Let $S(m)$ denote a star graph on $m+1$ nodes. We define {\it claw number} of a graph $G$ as the largest positive integer $k$ such that $S(k)$ is an induced subgraph of $G$ and denote it as $\claw$. Let $G$ be an AT-free graph with chromatic number $\chi(G)$ and claw number $\claw$. In this paper we will show that $\boxi(G) \leq \chi(G)$ and this bound is tight. We also show that $\cub(G) \leq \boxi(G)(\ceil{\log_2 \claw} +2)$ $\leq$ $\chi(G)(\ceil{\log_2 \claw} +2)$. If $G$ is an AT-free graph having girth at least 5 then $\boxi(G) \leq 2$ and therefore $\cub(G) \leq 2\ceil{\log_2 \claw} +4$.