# Questions tagged [embeddings]

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### Graphs such that every rotation system admits an embedding on a surface of small genus

Let $G$ be a finite, simple, undirected graph. What conditions on $G$ ensure that every rotation system of $G$ corresponds to a cellular embedding of $G$ on an oriented surface of small genus? (e.g. ...
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### Citation for isometric embeddability of $\ell_2$ into $\ell_p^\binom{n}{2}$ for $p \geq 1$?

I need to use the following well-known result in my paper: Let $X$ be a set of $n$ points in $\mathbb{R}^d$. Then $(X,\ell_2^d)$ embeds isometrically in $\ell_p^\binom{n}{2}$ for all $p \geq 1$. ...
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### Embedding points in Euclidean space into a box

If I give you a set of points in Euclidean space, is there a criterion to determine whether there exists a (potentially higher-dimensional) rectangular prism / box that has these points as their ...
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### Embedding a graph with specified edge lengths in d-dimensional space

Given any undirected edge-weighted graph (with weights > 0) and some dimension d, is there a way to assign positions in $\mathbb{R}^d$ to those vertices such that all of the edges between them have ...
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### Embedding of "large" graphs into random graph

Let $G \sim G_{n,p}$ be a binomial random graph and consider a sequence of graphs $\{H_n\}_n$. When $|H_n|=\mathcal{O}(1)$, then one knows the exact threshold $p_0$ for embeddabiliy of $H_n$ in $G$. ...
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### Are there Similar Distance Binary Error Correcting Codes?

I'm trying to find a low distortion embedding of the trivial metric space into hamming space. It seems this should be doable by using a large set of low dimensional vectors, with approximately equal ...
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### Embedding a graph in the euclidean space

Given a graph $G=(V,E)$, find a mapping $f\colon V \rightarrow \mathbb R^d$ such that for every edge $(u,v) \in E$ we have that $||f(u)-f(v)|| \leq r$; and for every $(u,v) \not \in E$, we have the ...
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### Book Embedding Duality of Graphs

The definition of Book Embedding on Wikipedia: "A book embedding is a generalization of planar embedding of a graph to embeddings into a book, a collection of half-planes all having the same line as ...
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### The random densification technique-JL lemma

In Ailon's paper (p.3): How $1/(20nd)$ is obtained?
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### Which properties of planar graphs generalize to higher dimension / hypergraphs?

A planar graph is a graph which can be embedded in the plane, without having crossing edges. Let $G=(X,E)$ be a $k$-uniform-hypergraph, i.e. an hypergraph such that all its hyperedges have size k. ...
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