# cube

## Edge-antipodal colorings of cubes ★★

Author(s): Norine

We let denote the -dimensional cube graph. A map is called edge-antipodal if whenever are antipodal edges.

Conjecture   If and is edge-antipodal, then there exist a pair of antipodal vertices which are joined by a monochromatic path.

Keywords: antipodal; cube; edge-coloring

## Simplexity of the n-cube ★★★

Author(s):

Question   What is the minimum cardinality of a decomposition of the -cube into -simplices?

Keywords: cube; decomposition; simplex

## Cube-Simplex conjecture ★★★

Author(s): Kalai

Conjecture   For every positive integer , there exists an integer so that every polytope of dimension has a -dimensional face which is either a simplex or is combinatorially isomorphic to a -dimensional cube.

Keywords: cube; facet; polytope; simplex