The chromatic number of 4-dimensional lattices

TL;DR

This paper determines the chromatic number of all 4-dimensional lattices by classifying Voronoi graphs, using combinatorial, geometric, and computational methods including SAT solvers.

Contribution

It provides a complete classification of the chromatic numbers for 4D lattices based on Voronoi graph analysis and introduces systematic methods for graph isomorphism checking.

Findings

Chromatic numbers for all 4D lattices are determined.

A systematic approach for checking Cayley graph isomorphism is developed.

SAT solvers are used to compute chromatic numbers of finite graphs.

Abstract

The chromatic number of a lattice in n-dimensional Euclidean space is defined as the chromatic number of its Voronoi graph. The Voronoi graph is the Cayley graph on the lattice having the strict Voronoi vectors as generators. In this paper we determine the chromatic number of all 4-dimensional lattices. To achieve this we use the known classification of 52 parallelohedra in dimension 4. These 52 geometric types yield 16 combinatorial types of relevant Voronoi graphs. We discuss a systematic approach to checking for isomorphism of Cayley graphs of lattices. Lower bounds for the chromatic number are obtained from choosing appropriate small finite induced subgraphs of the Voronoi graphs. Matching upper bounds are derived from periodic colorings. To determine the chromatic numbers of these finite graphs, we employ a SAT solver.