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22 votes
2 answers
900 views

Is every 1-million-connected graph rigid in 3D?

It is an old result that every $6$-connected graph is rigid in $\mathbb{R}^2$: Lovász, László, and Yechiam Yemini. "On generic rigidity in the plane." SIAM Journal on Algebraic Discrete ...
Joseph O'Rourke's user avatar
18 votes
2 answers
700 views

Can all unit-distance graphs have their vertices at algebraic integers?

A graph $G$ is described as a unit-distance graph if there exists a function $f:G \rightarrow \mathbb{C}$ such that for every edge $(u,v) \in E(G)$, we have $|f(u) - f(v)| = 1$. Obviously, we can ...
Adam P. Goucher's user avatar
13 votes
0 answers
751 views

$\epsilon$-nets with respect to the cut norm

The cut norm $||A||\_C$ of a real matrix $A = (a_{i,j}) \in \mathcal{R}^{n\times n}$ is the maximum over all $I \subseteq [n], J \subseteq [n]$ of the quantity $\left|\sum_{i \in I, j \in J}a_{i,j}\...
Aaron's user avatar
  • 794
11 votes
1 answer
369 views

The number of relevant scales for a finite metric space

For an $n$-element metric space $X=\{x_1,\dots,x_n\}$ with metric $d$ we introduce an array containing $\frac{n(n-1)}2$ numbers $d(x_i,x_j)$, $i<j$. We assume that all distances are at least $1$. ...
Mikhail Ostrovskii's user avatar
11 votes
3 answers
2k views

Could a perfect squared square be split into two perfect squared squares?

This is a geometric puzzle though it might conceivably also define a special class of Pythagorean triples. A perfect squared square PSS is a square (as a plane figure) partitioned into smaller ...
Mirko's user avatar
  • 1,375
11 votes
1 answer
506 views

"minimal" embedding of bipartite graphs on a sphere

Here is an easy to pose problem I've encountered (but haven't been able to solve or disprove): Let (V,E) be a bipartite graph with the following property – the girth of the graph (i.e. the length of ...
Izhar Oppenheim's user avatar
9 votes
3 answers
605 views

Separating points in the plane II

Let A be a set of $2m$ points on the plane so that no open set of diameter $2$ has more than m of them. Define $A+A+...+A$ ($k$ times) to be the multiset of $k$-sums from $A$. That is, we consider all ...
TOM's user avatar
  • 2,288
8 votes
1 answer
567 views

Joining the $2^k$ points of $\{0,1\}^k$ with the shortest tree

Let $k$ be a given positive integer, and then consider the unit hypercube $\{0, 1\}^k \subset \mathbb{R}^k$ (i.e., a $k$-dimensional "cube" in the well-known Euclidean space). We need to ...
Marco Ripà's user avatar
  • 1,451
7 votes
3 answers
474 views

Polygonal paths and polygons with prescribed set of vertices

Let $A$ be a finite set of points in the plane. How can we determine if there is a simple open polygonal path (i.e. without intersections), whose vertices are exactly $A$, with no straight angles ...
Algirdas Rugys's user avatar
7 votes
1 answer
153 views

Above/below directed graph on cells of arrangement of lines

This question concerns the structure of a directed graph built on the cells of an arrangement of lines. My basic question is whether this graph has been studied before, perhaps in another guise. I ...
Joseph O'Rourke's user avatar
6 votes
1 answer
483 views

Separating pairs of points in R^n

Let $A$ be a set of $2k$ points in $\mathbb{R}^n$ such that no open set in $\mathbb{R}^n$ of diameter $2$ contains more than $k$ of these points. What is the largest possible distance $r_n>0$ one ...
TOM's user avatar
  • 2,288
6 votes
2 answers
1k views

Minimum spanning tree of a weighted graph

I have a connected graph $G=(V,E)$ in $n$ vertices. The edge weights are non-negative and form a metric space, thus for vertices $u,v,w \in V$ , such that $(u,v), (v,w), (w,u)\in E$ we have $r(u,w) \...
MAKCL's user avatar
  • 89
6 votes
2 answers
268 views

Counting valid coordinates

We are given a matrix $D = (d(i,j))_{1 \leq i,j \leq n}$ such that $d(x,z) \leq d(x,y) + d(y,z)$ for each $1 \leq x,y,z \leq n$. It is also known that $d(x,y) \in \mathbb{N}$ (In this question $0 \in \...
real's user avatar
  • 323
5 votes
2 answers
237 views

Volume of the convex hull of the set of all graphic sequences of a given length

Consider the set of all graphic sequences with $n$ elements as a subset of $\mathbb{R}^{n}$, namely let $$D(n)=\{(d_{1},\dots,d_{n})\in\mathbb{Z}_{+}^{n}:d_{1}\geq\dots\geq d_{n},\ \sum_{i=1}^{n}d_{i}\...
Sergiy Kozerenko's user avatar
4 votes
1 answer
451 views

When can a 3-dimensional triangulation be isometricaly embedded in R^n?

Consider a triangulation of some bounded region of $R^3$ with a (finite) set of tetrahedra (like in Regge calculus). It can be thought of as a simplicial 3-complex with specified lengths of edges. The ...
nadbor's user avatar
  • 221
4 votes
1 answer
422 views

Can $n$ circles on a plane generate $m$ intersection points where at least $k$ circles intersect?

Can $n$ circles on a plane generate $m$ intersection points where at least $k$ circles intersect? For $k = 2$ the answer is obvious since we can always place circles so that every one of them ...
myro's user avatar
  • 63
4 votes
0 answers
66 views

Convergence of graph geodesics to geodesics on metric spaces

Let $(X,d)$ be a compact length space metric space $\mathbb{X}_{\delta}$ be a $\delta$-packing on $X$ and, for every $k\in \mathbb{N}_+$, let $G_{k,\delta}=(\mathbb{X}_{\delta},\mathcal{E}_k,W_k)$ ...
Math_Newbie's user avatar
4 votes
0 answers
94 views

Finding closest set of K disjoint hyperspheres to a point in $\mathbb{R}^n$ with uniform radius

I am interested in the following problem: in $\mathbb{R}^n$, we have $N$ overlapping hyperspheres all with the same radius. Given a point $p$ in $\mathbb{R}^n$, the objective is to find the $K$ non ...
eagle34's user avatar
  • 161
3 votes
3 answers
390 views

Can we uniquely define a graph to have the topology of a polytope via proper edge length selection?

I'll ask you to consider a situation wherein one has a series of edges for a graph, $(e_1, e_2, ..., e_N) \in E$, each with a specifiable length $(l_1, l_2, ..., l_N) \in L$, and the goal is to insure ...
ShallowBlue's user avatar
3 votes
1 answer
443 views

What is the expected value for this

If there are $8$ random points in the plane whose horizontal coordinate and vertical coordinate are uniformly distributed on the open interval $\left(0,1\right)$, what is the expected largest size of ...
user0o's user avatar
  • 31
3 votes
1 answer
179 views

When is a $k$-distance-transitive graph already distance-transitive?

Call a (finite and connected) graph $k$-distance-transitive if its symmetry group acts transitively on the pairs in each one of the sets $$D_\delta:=\{(i,j)\in V\times V\mid \mathrm d(i,j)=\delta\},\...
M. Winter's user avatar
  • 13.6k
3 votes
0 answers
134 views

Two questions on counterexamples to Borsuk's conjecture and ball-packings

In 1933 Karol Borsuk conjectured the following Can every bounded subset $E$ of $\mathbb{R}^d$ be partitioned into $(d+1)$ sets, each of which has a smaller diameter than $E$? Whilst new to this ...
Felix's user avatar
  • 31
3 votes
0 answers
59 views

2-complexes which are coarse-grained graphs

A polygonal complex $K$ is said to be geometrically 2-dimensional if the topological space it defines is a surface (boundaries are allowed). It is said to be $C$-quasi-1-dimensional (for some $C>0$)...
Itamar Vigdorovich's user avatar
2 votes
2 answers
255 views

What is the smallest number of subsets in such a subdivision?

Given any $30$ points in the plane, what is the smallest number of subsets in a subdivision of the set of $30$ points into subsets such that all the points in each subset are on the boundary of the ...
Diorn's user avatar
  • 21
2 votes
1 answer
138 views

Covering a circle with small balls centered at nodes on a tiling

In $\mathbb{R}$, we have $n$ finite sets, namely $\{A_1,A_2,\dots, A_n\}$. From them, we define a tiling: $$ T := \{x\in \mathbb{R}^n: \forall i \in \{1,2,\dots,n\}, ~x_i\in A_i\} = \prod_{i=1}^nA_i $$...
Brian's user avatar
  • 203
2 votes
1 answer
247 views

Are combinatorial configurations whose Levi graphs may be represented as covering graphs over voltage graphs realizable with pseudolines?

This question is related to this previous question. Many combinatorial configurations have Levi graphs which may be represented as derived graphs obtained from voltage graphs over a cyclic group; in a ...
Leah Wrenn Berman's user avatar
1 vote
1 answer
153 views

How many nodes does a ball of radius $r$ in the Johnson graph $J(n,k)$ contain?

1) How many nodes does a ball of radius $r$ in the Johnson graph $J(n,k)$ contain (Volume)? 2) How many nodes $v$ does a ball with center $x$ of radius $r$ in the Johnson graph $J(n,k)$ contain such ...
user avatar
1 vote
1 answer
119 views

Characterizing 1-ended graphs

I just came across the notion of ends of a space, and I wonder if the following are equivalent for $G$ a locally finite connected graph: There exists an infinite path $v_1,v_2,\dots$ in $G$ which ...
I. Haage's user avatar
  • 233
1 vote
1 answer
178 views

Planar eucliean bipartite matching with squared distances

This is probably a really stupid question, but suppose I have two sets of points in the plane $X$ and $Y$ each with cardinality $|X| = |Y| = n$. For any bipartite matching $M$ between $X$ and $Y$, ...
Mads Simonsen's user avatar
1 vote
1 answer
110 views

Distance pairs in labeled directed graph

Suppose we have a simple directed graph with $n$ nodes and $m$ edges, and we label each edge from $1$ to $m$ (with distinct labels). Define the weighted "length" of a directed path to be the maximum ...
Antimony's user avatar
  • 130
1 vote
0 answers
67 views

Conjecture on the increasing efficiency of the shortest minimum-link polygonal chains covering any grids of the form $\{0,1,2\}^k$ as $k$ grows

From the well-known Nine dots problem, we know that we need a polygonal chain with at least $4$ edges to connect the $9$ points of the planar grid $G_{3,2}:=\{\{0, 1, 2\} \times \{0, 1, 2\}\} \subset \...
Marco Ripà's user avatar
  • 1,451
0 votes
1 answer
229 views

Is this bounded?

May be better to ask for help here. Let $v_{1}$, $v_{2}$, $\ldots$, $v_{m}$ be the vertices of a convex polygon in the plane and $v_{m+1}$ be a vertex in the interior of the convex polygon. Connect ...
Palt's user avatar
  • 1
0 votes
0 answers
65 views

Maximal number of times distance $1$ can occur among $n$ points in the plane [duplicate]

For $n\in\mathbb N$, let $f(n)$ be the maximal number of times distance $1$ can occur among $n$ points in the plane: $$ f(n) = \max_{ \{ x_1,\ldots,x_n \} \subset \mathbb R^2} \# \big \{ i<j : \| ...
André Henriques's user avatar