cube

Edge-antipodal colorings of cubes ★★

Author(s): Norine

We let $Q_d$ denote the $d$-dimensional cube graph. A map $\phi : E(Q_d) \rightarrow \{0,1\}$ is called \emph{edge-antipodal} if $\phi(e) \neq \phi(e')$ whenever $e,e'$ are antipodal edges.

\begin{conjecture} If $d \ge 2$ and $\phi : E(Q_d) \rightarrow \{0,1\}$ is edge-antipodal, then there exist a pair of antipodal vertices $v,v' \in V(Q_d)$ which are joined by a monochromatic path. \end{conjecture}

Keywords: antipodal; cube; edge-coloring

Simplexity of the n-cube ★★★

Author(s):

\begin{question} What is the minimum cardinality of a decomposition of the $n$-cube into $n$-simplices? \end{question}

Keywords: cube; decomposition; simplex

Cube-Simplex conjecture ★★★

Author(s): Kalai

\begin{conjecture} For every positive integer $k$, there exists an integer $d$ so that every polytope of dimension $\ge d$ has a $k$-dimensional face which is either a simplex or is combinatorially isomorphic to a $k$-dimensional cube. \end{conjecture}

Keywords: cube; facet; polytope; simplex