On the Boxicity of Kneser Graphs and Complements of Line Graphs

TL;DR

This paper investigates the boxicity of Kneser graphs and complements of line graphs, providing bounds and exact values for specific cases, advancing understanding of their geometric representations.

Contribution

It establishes new upper and lower bounds for the boxicity of Kneser graphs and complements of line graphs, including exact values for certain parameters.

Findings

Derived a general upper bound for boxicity of Kneser graphs.

Established a lower bound matching the upper bound up to a constant factor.

Determined that boxicity of Kn(2,n) is either n-3 or n-2 for n ≥ 5.

Abstract

An axis-parallel $d$-dimensional box is a cartesian product $I_1\times I_2\times \dots \times I_b$ where $I_i$ is a closed sub-interval of the real line. For a graph $G = (V,E)$, the $boxicity \ of \ G$, denoted by $\text{box}(G)$, is the minimum dimension $d$ such that $G$ is the intersection graph of a family $(B_v)_{v\in V}$ of $d$-dimensional boxes in $\mathbb R^d$. Let $k$ and $n$ be two positive integers such that $n\geq 2k+1$. The $Kneser \ graph$ $Kn(k,n)$ is the graph with vertex set given by all subsets of $\{1,2,\dots,n\}$ of size $k$ where two vertices are adjacent if their corresponding $k$-sets are disjoint. In this note, we derive a general upper bound for $\text{box}(Kn(k,n))$, and a lower bound in the case $n\ge 2k^3-2k^2+1$, which matches the upper bound up to an additive factor of $\Theta(k^2)$. Our second contribution is to provide upper and lower bounds for the boxicity of the complement of the line graph of any graph $G$, and as a corollary, we derive that $\text{box}(Kn(2,n))\in \{n-3, n-2\}$ for every $n\ge 5$.