Cube tiling and covering a complete graph

Abstract

A collection of translates of an n-dimensional cube forms a tiling if the n-space is covered by its elements and no two cubes have a common interior point. If two n-dimensional cubes share a whole common (n−1)-dimensional face then they are called a twin pair. In 1930 Keller conjectured that in any n-dimensional cube tiling there is a twin pair. In 1982 Lawrence, using geometrical argumentation, gave a graph theoretical form for Keller's conjecture. We derive a slightly different graph test from its group theoretical version