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Combinatorics » Ramsey Theory

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 edge-antipodal if $ \phi(e) \neq \phi(e') $ whenever $ e,e' $ are antipodal edges.

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.

Keywords: antipodal; cube; edge-coloring

Posted by mdevos
updated August 20th, 2010
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Geometry

Simplexity of the n-cube ★★★

Author(s):

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

Keywords: cube; decomposition; simplex

Posted by mdevos
updated August 6th, 2008
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Geometry » Polytopes

Cube-Simplex conjecture ★★★

Author(s): Kalai

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.

Keywords: cube; facet; polytope; simplex

Posted by mdevos
updated May 9th, 2008
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