# Coloring the Voronoi tessellation of lattices

## TL;DR

This paper introduces the concept of the chromatic number for lattice Voronoi tessellations, computes it for various lattices, and explores its asymptotic behavior and spectral bounds, connecting geometry, algebra, and spectral graph theory.

## Contribution

It defines the lattice chromatic number, computes it for key lattices, and establishes spectral bounds, advancing understanding of lattice coloring and its asymptotics.

## Key findings

- Chromatic number computed for root lattices, duals, and Leech lattice.
- Spectral lower bounds for lattice chromatic number established.
- Asymptotic behavior of lattice chromatic number analyzed in high dimensions.

## Abstract

In this paper we define the chromatic number of a lattice: It is the least number of colors one needs to color the interiors of the cells of the Voronoi tessellation of a lattice so that no two cells sharing a facet are of the same color. We compute the chromatic number of the root lattices, their duals, and of the Leech lattice, we consider the chromatic number of lattices of Voronoi's first kind, and we investigate the asymptotic behaviour of the chromatic number of lattices when the dimension tends to infinity. We introduce a spectral lower bound for the chromatic number of lattices in spirit of Hoffman's bound for finite graphs. We compute this bound for the root lattices and relate it to the character theory of the corresponding Lie groups.