Coloring translates and homothets of a convex body

TL;DR

This paper improves bounds on the chromatic number of intersection graphs formed by translates and homothets of convex bodies, relating it linearly to the clique number in Euclidean space.

Contribution

It provides new upper and lower bounds on the chromatic number for these geometric intersection graphs, advancing understanding of their coloring properties.

Findings

Improved upper bounds on chromatic number.

New lower bounds on chromatic number.

Linear relation between chromatic number and clique number.

Abstract

We obtain improved upper bounds and new lower bounds on the chromatic number as a linear function of the clique number, for the intersection graphs (and their complements) of finite families of translates and homothets of a convex body in $\RR^n$.