All Questions
35 questions with no upvoted or accepted answers
16
votes
0
answers
1k
views
Optimal monotone families for the discrete isoperimetric inequality
Background: the discrete isoperimetric inequality
Start with a set $X=\{1,2,...,n\}$ of $n$ elements and the family $2^X$ of all subsets of $X$.
For a real number $p$ between zero and one, we consider ...
- 24.7k
14
votes
0
answers
1k
views
The threshold for a perfect matching in a random subgraph of a regular bipartite graph?
The following question seems very natural.
It is a well known consequence of Hall's Theorem that every regular bipartite graph has a perfect matching. Another classical result states that the ...
- 1,643
10
votes
0
answers
222
views
Asymptotics of subgraph densities in graphons
In Pittel (1989)'s solution to a problem of Knuth (1976) on the expected number of stable matchings between $n$ men and $n$ women under uniform random preferences, it was shown that, as $n \to \infty$,...
- 986
8
votes
0
answers
304
views
"Meritocratic" pyramid schemes
There have been a couple of times in my life when people from multi-level marketing organizations attempted to recruit me. I listened to what they had to say, and both times I did not get involved ...
- 2,075
8
votes
0
answers
181
views
Self-avoiding walks on strips
A strip is a locally finite graph which admits a quasi-transitive (i.e. finitley many orbits on vertices) action of $\mathbb Z$. A self avoiding walk is a walk which visits no vertex more than once.
...
- 2,671
7
votes
0
answers
171
views
What is known about the distribution of lengths of the cycle you get by adding an edge to a uniform spanning tree?
Let $G$ be a finite, connected graph. Let $T$ be a uniform spanning tree, and let $e$ be a uniformly random edge not in $T$. When we add $e$ to $T$, we get a subgraph with a unique cycle, $C$. I am ...
- 1,462
6
votes
0
answers
164
views
Hamilton cycles in random graphs with just enough connectivity
What is the asymptotic probability that $G$ has a Hamilton cycle if $G$ is a random $n$ vertex $\frac{4}{3}n$ edge graph, with minimum degree 2 and without degree 2 vertices at distance 1 or 2 to each ...
- 7,545
6
votes
0
answers
172
views
Uniformly sampling from the set of all simplicial maps
Let $K$ and $L$ be finite simplicial complexes that remain fixed throughout.
How does one efficiently sample (according to the uniform distribution) elements from the finite set of simplicial maps ...
- 15.5k
4
votes
0
answers
1k
views
Number of arrangements that contain at least 1 path from top to bottom of 2D matrix
I have a $n\times n$ matrix of objects. $n'$ objects are black, and the rest $n^2-n'$ are white.
With that information, I can easily calculate the total number of black element arrangements that exist ...
- 47
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 ...
- 161
4
votes
0
answers
220
views
Navigation in a graph
The problem
Let $G=(V,E)$ be a graph. $k = O\left(\log(|V|)\right)$ distinct vertices are picked randomly from $V$. We call the set of chosen $k$ vertices $T$.
Assumptions about the graph: You may ...
- 323
4
votes
0
answers
580
views
Monotonic properties of harmonic functions on graphs
I have a question concerning monotonic properties of "generalized harmonic functions" on graphs. I am a physicist and I didn't take any separate courses in neither graph theory nor discrete harmonic ...
- 965
3
votes
0
answers
81
views
Can we remove the restriction on a parameter in Talagrand concentration inequality?
Recently I am trying to use Talagrand concentration inequality to do something on graphs. I find a version from the book of Molloy and Reed ''Graph Colouring and Probabilistics Method''. I attached a ...
- 1,190
3
votes
0
answers
190
views
Probabilistic optimization problem on tree vertex selection without replacement proportional to the degree
We are given a tree $T(V,E)$ with $|V|=n$ vertices, where $V=\{v_1,v_2,\ldots, v_n\}$. We denote by $d_i$ the degree of vertex $v_i$ for all $i\in\{1,2,\ldots,n\}$.
In a sequential fashion, we select ...
- 1,677
2
votes
0
answers
83
views
Zero-One law for Hamiltonian path subgraphs of Hamming Distance Graphs?
$(\alpha,\beta,d)$-Hamming Distance Graph $G_d(\alpha,\beta)$ for $\alpha,\beta\in(0,1]$ is a graph on $2^d$ vertices $v_0,\dots,v_{2^d-1}$ with edges $(v_i,v_j)\in\mathcal E(G_d)$ iff $0<\sum_{t=1}...
- 13.9k
2
votes
0
answers
83
views
Asymptotic results in unbalanced left $d$-regular expander graphs
Let $U = [n]$ and $V = [m]$ be sets of nodes with $n > m$ and $E = U\times V$ be a set of edges. Let $\mathcal{N}(S)$ be the set of neighbors of a subset $S$ from $U$ or $V$.
Call a graph $G = (U, ...
- 409
1
vote
0
answers
72
views
How to understand "sparse graph limits"
For an $n$-vertex graph $G$, we say it is a sparse graph if $e(G)=o(n^2)$. Otherwise if $e(G)=\theta (n^2)$, we say it is a dense graph.
For a sequence of dense graphs $G_1,G_2,\dots,$ we know that it ...
- 349
1
vote
0
answers
134
views
Number of ways to place 4 kings on nxn chessboard
I have a $n\times n$ chessboard and 4 kings inside it. My goal is to count the number of arrangements where some of them are non-attacking or mutually attacking, for example:
In the case where the $4$...
- 47
1
vote
0
answers
84
views
Percolative process distribution not equivalent to coupon collector problem distribution
I have a process where; given a $n\times 1$ matrix initially empty, an element is inserted in it at a random position, with the possibility of repeating the insertion at a filled cell. Then, after a ...
- 47
1
vote
0
answers
62
views
Nonintersecting witnesses of connectivity events in graphs
In my research I stumbled across a following result:
Let $G = (V, E)$ be a multigraph with three chosen vertices $a, b, c \in V$. We color its edges into red and blue colors: $E = R \sqcup S$. Events ...
1
vote
0
answers
60
views
Correct dependence for "Local Coloring"
In Alon-Spencer's book, Probabilistic Lens #8, it is proven that for each $k$, there exists $\epsilon = \epsilon(k)>0$ such that for all large $n$, there exists an $n$-vertex graph $G$ with ...
- 3,499
1
vote
0
answers
127
views
Delocalization of eigenvectors of graph Laplacians
Let $(V,E)$ be an undirected, connected graph with $n$ nodes. The graph Laplacian is defined as $L = D - A$, where $D$ is the degree matrix and $A$ is the adjacency matrix. Let $0 = \lambda_1 < \...
- 41
1
vote
0
answers
140
views
Count shortest path with different lengths in random graph
Let $G(n,p)$ be an Erdos-Renyi random graph on $n$ vertices with probability $p$, i.e. for each pair of vertices, they are connected directly by an undirected edge with probability $p$. Suppose we are ...
- 1,495
1
vote
0
answers
54
views
Age of the most recent common ancestor for the neutral Wright-Fisher model
The neutral Wright-Fisher model with $n$ individuals is a genealogical model often used in population genetics that can be described as follows: at all generations, there are exactly $n$ individuals, ...
- 11
1
vote
0
answers
61
views
What is the minimal $m$ for which the independence graph is $n$-universal?
Suppose, an $m$ sided die is rolled. Let's define the independence graph $I_m$ as a graph with the set of all possible events as vertices, and edges between two events iff they are independent.
...
- 2,618
1
vote
0
answers
117
views
Entropy of endpoints of a random walk in a dense graph
Let $p\in[0,1]$ be a constant and let $G$ be a graph with $n$ vertices and $\approx p\binom{n}{2}$ edges. If you'd like, consider $p=1/2$.
Let $X$ be a random vertex of $G$ chosen proportional to ...
- 761
1
vote
0
answers
87
views
How to estimate the size of balanced biclique in random bipartite graph?
We have a random bipartite graph $G=(V,U,E)$ and $|V|=|U|=n$, in which any vertex pair $<v,u>$ ($v\in V$,$u\in U$) exists an edge with probability $p$. A balanced bipartite complete graph is a ...
- 11
1
vote
0
answers
255
views
Multiple Bipartite graphs and matchings
I've been told recently that it's better i just for help regarding my 'specific' problem rather than lots of little questions around the same topic which appear somewhat unclear. I would first like to ...
- 903
1
vote
0
answers
46
views
Is there an effective algorithm for finding "minimal discovery times" for large graphs?
Consider a large, probably sparse graph with Markovian random walkers on it.
Define the discovery time as the expected time to first reach a vertex by
random walk from a uniform start. Are there ...
- 355
0
votes
0
answers
63
views
Arrangements of fixed $k$-polyplets in a $n\times n$ matrix
Recently, I asked a question about the number of arrangements of $k$ elements inside a $n\times n$ matrix with certain restrictions. The one I´m actually interested in for this question is in its 2. ...
- 47
0
votes
0
answers
55
views
Counting matrix paths for (n,m>2) matrices
Given a $n\times m$ matrix with $k$ elements inside it, I need to calculate the number of arrangements of those $k$ elements that form at least 1 path from the top to bottom matrix row composed of the ...
- 47
0
votes
0
answers
133
views
is there an example in planar graph that using probabilistic methods
The probabilistic method is a technique for proving the
existence of an object with certain properties by showing that
a random object chosen from an appropriate probability
distribution has the ...
- 1,851
0
votes
0
answers
39
views
hypergraph product that preserve expansion properties
I am looking for a hypergraphs product of hypergraph H1,H2 that preserves some expansion properties of H1,H2.
The expansion property I am looking at is HD-random walk.
The product I am looking for is ...
- 340
0
votes
0
answers
72
views
A random variable standing for the size of connected component including a given node in a tree
Suppose we have a tree $T = (V,E)$, in which each nodes $v_i \in V$ has a probability $p_i$ to vanish. Let $v_0\in V$, we define random variable $\boldsymbol{X} = \boldsymbol{X}(T, v_0)$ stands for ...
- 1,551
0
votes
0
answers
216
views
Computation on Random Bipartite graphs
I'm looking at a random bipartite graph $K_{\omega(n)}*K_{\omega(n)}$ where $\mathrm{log}(n)\leq \omega(n) \leq n^{1/2}$, in which each of the $\omega(n)^{2}$ edges is placed randomly with probability ...
- 903