# Questions tagged [hypercube]

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

**14**

votes

**0**answers

367 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 ...

**12**

votes

**4**answers

718 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 $(0,0,\ldots,0)$ ...

**5**

votes

**0**answers

195 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,
...

**3**

votes

**1**answer

121 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{...