# The Resolution of Keller’s Conjecture

@article{Brakensiek2020TheRO,
title={The Resolution of Keller’s Conjecture},
author={Joshua Brakensiek and Marijn J. H. Heule and John Mackey},
journal={Automated Reasoning},
year={2020},
volume={12166},
pages={48 - 65}
}
• Published 9 October 2019
• Materials Science
• Automated Reasoning
We consider three graphs, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_{7,3}$$\end{document}, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin…

### Gluing and cutting of cube tiling codes in dimension six

Let $S$ be a set of arbitrary objects, and let $s\mapsto s'$ be a permutation of $S$ such that $s''=(s')'=s$ and $s'\neq s$. Let $S^d=\{v_1...v_d\colon v_i\in S\}$. Two words $v,w\in S^d$ are

### A Universal Construction for Unique Sink Orientations

• Computer Science
• 2022
This work was inspired by techniques from cube tilings of space and expands upon existing techniques in the area to develop generalized rewriting rules for USOs, a new construction framework which can be applied to all USOs.

### Geometry of Rounding

• Computer Science, Mathematics
ArXiv
• 2022
This work proves that for every d ∈ N, there is an explicit hypercube partition of R with k = d + 1 and = 1 2d, and investigates the optimality of the parameter and proves that any partition in this broad class of “reasonable” partitions including hypercube partitions must have ≤ 1 2 √ d.

### Filling space with hypercubes of two sizes – The pythagorean tiling in higher dimensions

We construct a unilateral lattice tiling of Rn$\mathbb {R}^n$ into hypercubes of two differnet side lengths p or q. This generalizes the Pythagorean tiling in R2$\mathbb {R}^2$ . We also show that

### Filling space with hypercubes of two sizes -- The pythagorean tiling in higher dimensions

We construct a unilateral lattice tiling of R n into hypercubes of two diﬀernet side lengths p or q . This generalizes the Pythagorean tiling in R 2 . We also show that this tiling is unique up to

### Square coloring planar graphs with automatic discharging

• Computer Science, Mathematics
ArXiv
• 2022
This paper uses a Linear Programming approach to automatically look for a discharging proof, and makes some progress towards Wegner’s conjecture for distance- 2 coloring of planar graphs, by showing that 12 colors arecient to color at distance 2 every planar graph with maximum degree 4.

### Machine Learning Methods in Solving the Boolean Satisfiability Problem

• Computer Science
ArXiv
• 2022
The evolving ML-SAT solvers from naive classifiers with handcrafted features to the emerging end-to-end SAT solvers such as NeuroSAT are examined, as well as recent progress on combinations of existing CDCL and local search solvers with machine learning methods.

### An attack on Zarankiewicz's problem through SAT solving

The Zarankiewicz function gives, for a chosen matrix and minor size, the maximum number of ones in a binary matrix not containing an all-one minor. Tables of this function for small arguments have

### Too much information: CDCL solvers need to forget and perform restarts

• Tom Krüger
• Computer Science
ArXiv
• 2022
It is demonstrated that clause learning (without being able to get rid of some clauses) can not only improve the runtime but can oftentimes deteriorate it dramatically, and it is found that the runtime distributions of CDCL solvers are multimodal.

### Too much information: Why CDCL solvers need to forget learned clauses

• Computer Science
PloS one
• 2022
It is demonstrated that clause learning (without being able to get rid of some clauses) can not only help the solver but can oftentimes deteriorate the solution process dramatically, and that the runtime distributions of CDCL solvers are multimodal.

## References

SHOWING 1-10 OF 34 REFERENCES

### Towards resolving Keller’s cube tiling conjecture in dimension seven

Abstract A cube tiling of ℝd is a family of pairwise disjoint cubes [0, 1)d + T = {[0, 1)d + t: t ∈ T} such that ∪t∈T([0, 1)d + t) = ℝd. Two cubes [0, 1)d + t, [0, 1)d + s are called a twin pair if

### Über die lückenlose Erfüllung des Raumes mit Würfeln.

Inhaltsverzeichnis. Einleitung 231 Vorbemerkungen 233 I. Teil: Eigenschaften der Raumerfüllung in n Dimensionen 233—241 § 1. Der Strahlensatz 233 § 2. Staffeln 234 § 3. Gerade und ungerade

### Extended Keller Graph and its properties

In the paper extended Keller graph Γ 3 d is defined and some of its properties, such as Hamiltonian, the independence number, the chromatic number, etc.,are proved. Moreover, the size of a maximum

### A combinatorial approach for Keller's conjecture

• Mathematics
• 1990
The statement, that in a tiling by translates of ann-dimensional cube there are two cubes having common (n-1)-dimensional faces, is known as Keller's conjecture. We shall prove that there is a

### A Cube Tiling of Dimension Eight with No Facesharing

A cube tiling of eight-dimensional space in which no pair of cubes share a complete common seven-dimensional face is constructed shows that the first dimension in which such a tiling can exist is seven or eight.

### A reduction of Keller's conjecture

A family of translates of a closedn-dimensional cube is called a cube tiling if the union of the cubes is the wholen-space and their interiors are disjoint. According to a famous unsolved conjecture