All Questions
13 questions
7
votes
1
answer
760
views
Difference Sets
Suppose
$$
P \subseteq \{1,2,\dots,N\},\quad |P| = K
$$
We calculate the differences as: $$d=p_i-p_j\mod N,\quad i\ne j$$
Now let $a_d$ denote the number of occurrence of $d$ (for $d = 1, 2, \dots , N ...
- 123
10
votes
1
answer
1k
views
How can we find n points on a plane so that as many pairs of points as possible have the same distance?
There are $n$ points on the plane, and we need to maximize the number of pairs of points which have the same Euclidean distance.
- 159
18
votes
1
answer
1k
views
Sperner's Lemma implies Tucker's Lemma - simple combinatorial proof
Sperner’s Lemma is often called the "combinatorial analog" of Brouwer’s Fixed Point Theorem, and similarly Tucker’s Lemma is often called the combinatorial analog of Borsuk–Ulam’s Theorem.
We can ...
- 6,917
6
votes
0
answers
657
views
Unique domino tiling
Question: how does one enumerate all star-convex $2n$-vertex sublattices of the plane that have the unique domino-tiling property?
Definitions:
A subset $S$ of the $xy$-plane is star-convex if there ...
- 1,090
16
votes
4
answers
1k
views
Squaring a square and discrete Ricci flow
Is this a theorem?
Every $3$-connected planar graph $G$ may be represented as
a tiling of a square by squares,
one square per node of $G$, with nodes connected in $G$
corresponding to tangent squares....
- 151k
9
votes
2
answers
484
views
Connected geometric thickness two
A graph $G = (V,E)$ has geometric thickness two if there exists an embedding $\varphi: V \rightarrow \mathbb{R}^2$ and a decomposition $E = E_1\cup E_2$ such that $G_1 = (V,E_1)$ and $G_2 = (V,E_2)$ ...
- 479
6
votes
1
answer
122
views
Have this generalization of Indifference graphs been studied before?
Indifference graphs are those graphs $G=(V,E)$ for which there exists a real-valued function $f$ defined on $V(G)$ such that, if $u$ and $v$ are distinct vertices, $|f(u)−f(v)| \lt 1$ if and only if $\...
- 519
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 ...
- 63
4
votes
2
answers
512
views
Geometry, Number Theory and Graph Theory of n-gon, permutation and graph labeling?
Given $n$ and $t$ lengths $ l_i, 1\leq l_1<l_2<\cdots<l_t\leq n-1$, of directed diagonals within an $n$-gon such that $l_1+\cdots+l_t\neq 0 \pmod n)$. Does it exist a directed path within ...
- 221
3
votes
1
answer
157
views
Structure of boundary labelling in Sperner‘s Lemma
Consider a triangulated polygon in the 2-dimensional plane, where each vertex is labelled green, blue, or orange. Sperner's Lemma asserts that a fully-colored triangle exists in the triangulation, if ...
- 6,917
2
votes
0
answers
51
views
Regular graphs of tangent spheres
Problem 1. Let several non-overlapping spheres in $\mathbb{R^3}$ are given. For which $n$ it is possible that each sphere is tangent to exactly $n$ other spheres?
Consider the smallest sphere. Since ...
- 691
1
vote
1
answer
115
views
Bounds on lengths of boxes in bounded-degree box graphs
$\DeclareMathOperator{\box}{\operatorname{box}}$$\DeclareMathOperator{\cub}{\operatorname{cub}}$
This is a follow up and an extension of another question I asked recently.
A box graph is a graph ...
- 277
1
vote
1
answer
194
views
Bounds on lengths of intervals in bounded-degree interval graphs
A graph is said to be an interval graph if its vertices can be associated with (closed) intervals on the real line $\mathbb R$ and there is an edge between two vertices if and only if the ...
- 277