A note on lower bounds for boxicity of graphs

TL;DR

This paper introduces fractional boxicity as a linear relaxation-based lower bound for the boxicity of graphs, compares it with existing bounds, and emphasizes the importance of accuracy over simplicity in these bounds.

Contribution

It defines fractional boxicity, establishes its relation to existing bounds, and discusses the focus on accuracy for lower bounds of boxicity.

Findings

Fractional boxicity is at least as large as previous lower bounds.

A new natural lower bound for fractional boxicity is proposed.

The paper emphasizes the importance of accuracy in lower bounds for boxicity.

Abstract

The boxicity of a graph $G$ is the minimum non-negative integer $k$ such that $G$ can be isomorphic to the intersection graph of a family of boxes in Euclidean $k$-space, where a box in Euclidean $k$-space is the Cartesian product of $k$ closed intervals on the real line. In this note, we define the fractional boxicity of a graph as the optimum value of the linear relaxation of a covering problem with respect to boxicity, which gives a lower bound for its boxicity. We show that the fractional boxicity of a graph is at least the lower bounds for boxicity given by Adiga et al. in 2014. We also present a natural lower bound for fractional boxicity of graphs. Moreover we discuss and focus on "accuracy" rather than "simplicity" of these lower bounds for boxicity as the next step in Adiga's work.