# Questions tagged [hypercube]

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

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### Some questions about induced subgraphs of the directed hypercube graph

Let $Q^n$ be the hypercube graph in $n$ dimensions. Hao Huang famously showed that any induced subgraph on more than $2^{n-1}$ must have maximum degree $\geq \sqrt{n}$. It is also known that this ...
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1 vote
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### Maximize trace of precision matrix

Let Q be the uniform distribution over the hypercube $\{1, -1\}^{d}$. Let P be any distribution that has support including the hypercube. Define $\Sigma=\mathbb{E}_{x \sim P}[x x^\top]$. We'd like to ...
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1 vote
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### Hypercube and affine space [closed]

We have an affine subspace $A$ of dimension $d_{A}$ and a hypercube $C$ of dimension $d_{C}$ ($d_{C}=n$), with $d_{A}\lt d_{C}$ and both belong to $\Re ^{n}$. Each face of dimension $d_{C}-1$ of $C$ ...
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### 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 ...
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### Cycles in the hyperoctahedral group (symmetries of the hypercube)

Let $B_n$ be the hyperoctahedral group (the isometries of the $n$-dimensional hypercube). Let $k <n$ and consider the action of $B_n$ on the $k$-dimensional faces of the hypercube. What can you ...
143 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 $V_n$ ...
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### 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, ...
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### 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 $(0,0,\ldots,0)$ ...
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