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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.
![$ d \ge 2 $](/files/tex/11eec08cccba2fc15cb1ab0db8568a42a91cced5.png)
![$ \phi : E(Q_d) \rightarrow \{0,1\} $](/files/tex/685e9db47a7cdebc035cdddc5627c69a11d93982.png)
![$ v,v' \in V(Q_d) $](/files/tex/7bdec5e833c3f44aaec5ec53ad5017b232be6354.png)
Keywords: antipodal; cube; edge-coloring
Simplexity of the n-cube ★★★
Author(s):
Question What is the minimum cardinality of a decomposition of the
-cube into
-simplices?
![$ n $](/files/tex/ec63d7020a64c039d5f6703b8fa3ab7393358b5b.png)
![$ n $](/files/tex/ec63d7020a64c039d5f6703b8fa3ab7393358b5b.png)
Keywords: cube; decomposition; simplex
Cube-Simplex conjecture ★★★
Author(s): Kalai
Conjecture For every positive integer
, there exists an integer
so that every polytope of dimension
has a
-dimensional face which is either a simplex or is combinatorially isomorphic to a
-dimensional cube.
![$ k $](/files/tex/c450c3185f7285cfa0b88d3a903c54f7df601201.png)
![$ d $](/files/tex/aeba4a4076fc495e8b5df04d874f2911a838883a.png)
![$ \ge d $](/files/tex/616a25c014115485d010b8d180f072cc22ed1c53.png)
![$ k $](/files/tex/c450c3185f7285cfa0b88d3a903c54f7df601201.png)
![$ k $](/files/tex/c450c3185f7285cfa0b88d3a903c54f7df601201.png)
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