Timeline for Orthogonal representations of graphs
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 13, 2017 at 10:59 | comment | added | Fedor Petrov | In dimension 3 or more, I would try the following example: consider directions of all integer vectors in $[-N,N]^d$ for large $N$. Many of them are orthogonal, and it looks quite a rigid construction, that is, removing one edge from the orthogonality graph we should still have the same configuration. | |
| Oct 13, 2017 at 10:16 | comment | added | Peter Heinig | [...] the current OP with this alternative formulation. Explicitly, you could write: "Equivalently, let a $d$-dimensional f.o.r. of a graph $G=(n,E)$ mean an $n$-set $\{L_0,\dotsc,L_{n-1}\}$ of lines through the origin in $\mathbb{R}^d$ such that $ij\in E\ \Leftrightarrow\ $ $L_i\perp L_j$. (Here, $\perp$ means 'orthogonal')" (I could not escape my preference for indexing from $0$ to $n-1$ here; the main advantage here is the brevity with which one can make the problem statement precise by writing $G=(n,E)$, where $n$ denotes the finite ordinal $\{0,1,\dotsc,n-1\}$. Just a suggestion.) | |
| Oct 13, 2017 at 10:11 | comment | added | Peter Heinig | It seems useful to point out: because of the interestingly unusual (compared to some relevant articles) additional condition $\lvert\langle v_i,v_j\rangle\rvert=1\to i=j$ in the OP, it seems that a more appropriate mathematical concept to use is lines through the origin in $\mathbb{R}^d$. I.e., the suggestion is to represent vertices by lines through the origin, whereupon (this is the point) the condition of distinctness automatically implies the additional condition of non-parallelness. However, I would not recommend replacing the current formulation by this, only augmenting [...] | |
| Oct 12, 2017 at 21:21 | history | edited | pizzazz | CC BY-SA 3.0 |
added 4 characters in body
|
| Oct 12, 2017 at 15:05 | history | edited | pizzazz | CC BY-SA 3.0 |
added 225 characters in body
|
| Oct 12, 2017 at 14:49 | history | edited | pizzazz | CC BY-SA 3.0 |
added 10 characters in body
|
| Oct 10, 2017 at 18:56 | answer | added | Ivan Izmestiev | timeline score: 4 | |
| Oct 10, 2017 at 18:48 | answer | added | Peter Heinig | timeline score: 3 | |
| Oct 10, 2017 at 18:36 | history | edited | Peter Heinig | CC BY-SA 3.0 |
Purely stylistic improvements. Writing $uv$ for an edge of an undirected graphs (such as the graph in the OP) is a usual notation, but writing $(u,v)$, as in the initial OP, is not a usual notation, and wrong: there's no formalization of undirected graphs w.r.t. which edges were ordered pairs.
|
| Oct 10, 2017 at 17:31 | review | First posts | |||
| Oct 10, 2017 at 17:32 | |||||
| Oct 10, 2017 at 17:28 | history | asked | pizzazz | CC BY-SA 3.0 |