Questions tagged [hypercube]

For questions involving cubes in higher dimensions or the hypercube graphs.

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25 votes
1 answer
2k views

Number of hypercube unfoldings

While writing the code for this answer, I noticed that I not only could calculate the number of unfoldings of the $4$-cube, but also the number of the $n$-cube for more values of $n$. Basically, we ...
Moritz Firsching's user avatar
24 votes
1 answer
1k views

Which unfoldings of the $d$-dimensional hypercube tile $(d{-}1)$-space?

A six year old question, Which unfoldings of the hypercube tile $3$-space?, has just been answered by Moritz Firsching: All $261$ unfoldings tile space! So now we know: For $d=2$, the unfolding of ...
Joseph O'Rourke's user avatar
14 votes
0 answers
410 views

Monotone embedding of complete binary tree in hypercube

Embedding different graphs, especially binary trees, in the hypercube has a huge literature. However, I could not find anything if we restrict the embedding to be monotone. So I would like to ...
domotorp's user avatar
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12 votes
4 answers
2k views

Longest path through hypercube corners

Is the longest Hamiltonian path through the $2^d$ unit hypercube vertices known, where path length is measured by Euclidean distance in $\mathbb{R}^d$? The unit hypercube spans from $(0,0,\ldots,0)$ ...
Joseph O'Rourke's user avatar
5 votes
0 answers
483 views

Longest simple path through hypercube corners

This is a variation on a previously answered question, Longest path through hypercube corners. Here I am seeking the longest simple (non-self-intersecting) path through the unit hypercube's vertices, ...
Joseph O'Rourke's user avatar
3 votes
1 answer
156 views

Distance relation among points in high-dimensional hypercubes

Let $Q_{4n-1}$ be a unit hypercube of dimension $4n-1$. Has the following statement been proven? There are $4n$ vertices in $Q_{4n-1}$ such that the distance between each pair of them is $2\sqrt{...
C.F.G's user avatar
  • 4,165