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Questions tagged [hypercube]

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

14
votes
0answers
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 ...
5
votes
0answers
193 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, ...
4
votes
0answers
326 views

Hypercube edge-coloring problem

Question: Is there a pairing (a fixed point free involution) of the vertices of the $n$-dimensional cube graph, and a $2$-coloring of its edges such that the number of color changes needed to get from ...
3
votes
0answers
111 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 ...
2
votes
0answers
154 views

A question of Ahlswede and Katona: known lower bounds on $\beta(d,n)$?

Given a set $S\subseteq \{0,1\}^d$ of the Boolean hypercube of dimension $d$, define the average distance of $S$ as $$ \bar{d}(S) = \frac{1}{\lvert S\rvert^2} \sum_{x,y\in S} d_H(x,y)\tag{1} $$ where $...
2
votes
0answers
48 views

Hamiltonian cycle polytope for the hypercube graph

Let $Q_n$ denote the $n$ dimensional hypercube graph (i.e., graph formed from the vertices and edges of an n-dimensional hypercube). Denote the set of edges and vertices of $Q_n$ by $E_n$ and $V_n$ ...
2
votes
0answers
83 views

Counting Cycle Vertex Covers on Hypercube

Let $Q_n$ be the $n$-dimensional hypercube graph. How many vertex cycle covers exist on $Q_n$? (Presumably the best we can hope for are upper and lower bounds.) To be clear, a single "vertex cycle ...
1
vote
0answers
80 views

Characterizing normal vectors of affinely independent subsets of the hypercube

Suppose we are looking at the hypercube in $\mathbb{R}^n$. I tend to the think of the cube with vertex coordinates 0 or 1, but maybe this is easier for the $\pm1$ cube. Now suppose we have an affine ...
0
votes
0answers
141 views

Is there any result on the homomorphic images of hypercube graphs?

Let $Q_n$ be a hypercube graph and $\phi: Q_n\to G$ a surjective simplicial graph morphism i.e. if $u,v$ are adjacent vertices in $Q_n$ then either $\phi(u)=\phi(v)$ or $\phi(u),\phi(v)$ are adjacent. ...
0
votes
0answers
82 views

Commutator subgroup of rotational symmetries of the hypercube

I would like to know which is the commutator subgroup of the group of rotational isometries of the $n$-dimensional cube. The group i am talking about is the subgroup of $\{ \pm 1 \} \wr S_n$ ...
0
votes
0answers
130 views

Symmetries higher dimensional cube fixing subcubes

Which is the group of rotational isometries of an $n$ dimensional hypercube fixing an $m$ dimensional element (an $m$ dimensional subcube)? I know for example that it is $A_n$ for $m=0$ (symmetries ...