All Questions
5,777 questions
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Graph classes which have small edge k-cuts
I am interested in graph classes that have the following property: There exists a function $f(k)$ such that for every graph $G$ in the class, for every choice of $k$ vertices $v_1, \ldots, v_k$ in the ...
- 516
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0
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24
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Can both conditions about vertex degrees hold true in a planar graph?
I am working on a problem about planar graphs and trying to understand if two statements can both be true at the same time.
The problem states that for any planar graph with at least 3 or more ...
- 1
1
vote
0
answers
28
views
Integral hull of a polyhedron Q is polyhedron
Let $Q \subseteq R^n$ be a rational polyhedron and let $Q_I=Convexhull(Q \cap Z^n)$. By finite basis theorem, we have $Q=P+C$ for some rational polytope $P$ and finitely generated cone $C$ where $C=R_+...
- 21
-1
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0
answers
41
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Is it possible to backtrack an optimization solver? [closed]
I have an optimization problem and was using a linear programming optimizer to find solutions. However, I find that past a certain size, the problem becomes "infeasible" and has no solutions....
1
vote
0
answers
59
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Bipartite Representation of a Directed Graph
I am working on a combinatorial optimization problem and have constructed a bipartite graph as a representation of a directed graph.
The construction is as follows:
Given an initial directed graph $G$ ...
7
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0
answers
219
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Is there a Cayley graph with end space infinite and discrete?
A Cayley graph of a finitely generated group must be locally finite, and we know end spaces of locally finite graphs must be compact - so we can't have an infinite and discrete end space in this ...
- 407
5
votes
3
answers
274
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The max-clique chromatic number of a graph
Let $G = (V,E)$ be a graph. Every clique, that is, complete subgraph, is
contained in a maximal clique with respect to $\subseteq$ (this is
an easy consequence of Zorn's Lemma). Let $\newcommand{\MC}{\...
- 48.8k
1
vote
0
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122
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Growth polynomial of the Associahedron graph ? (Is it approximately Gaussian ?)
Consider Associahedron, consider graph build from its vertices and edges. Choose some vertex. Let us count the number of vertices on distances $k$ from the selected vertex. Write a generating ...
- 24.8k
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0
answers
34
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separator and vertex-connectivity
A definition of "separator" is the following: Let $G$ is an $n$-vertex graph, then $S\subseteq V(G)$ is a separator if there is a partition $V=A\cup B\cup S$ such that $|A|,|B|\le 2n/3$ and ...
- 281
3
votes
1
answer
131
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Is a simply connected locally 2-connected complex a union of spheres and planes?
Let $X$ be a (potentially infinite) 2-dimensional simplicial complex. Then each link at a vertex $x\in X$ is a graph.
Question. If $X$ is simply connected and each link is 2-connected (in the sense ...
- 13.6k
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0
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21
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Easy instance of set cover
I am trying to prove that a natural greedy algorithm solves the following instance of the set cover problem: for a set of elements $e\in U$ with a set of weights $w_e$, we define the cost of a subset ...
- 4,049
1
vote
1
answer
80
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What are the efficient algorithms to compute Hamiltonian paths on Cayley graphs of finite groups ? Can GAP do it?
The famous Lovasz conjecture predicts existence of the Hamiltonian path on Cayley graphs. In general finding such a path is NP-complete problem, but there are many heuristic algorithms.
Question 1: ...
- 24.8k
3
votes
1
answer
200
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Matrix-tree theorem for inverse matrices
Let $L$ be the Laplacian of a directed weighted graph on $n$ nodes, e.g., for $n=4$:
$$
L = \left(\begin{array}{cccc} w_{1,1}+w_{1,2}+w_{1,3}+w_{1,4} & -w_{1,2} & -w_{1,3} & -w_{1,4}\\ ...
- 20.2k
2
votes
1
answer
111
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Is there a ternary Cayley graph on 27 vertices that is a non-complete core?
Is there a non-complete ternary Cayley graph that is a core with $3^3 = 27$ vertices?
By a ternary Cayley graph, I mean a (simple, undirected) graph whose vertex set is $\mathbb{Z}_3^n := \bigoplus_{i ...
- 331
1
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0
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233
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Claimed proofs of graph labelling conjectures [closed]
The following recent series of arXiv papers claims to prove several of the most famous graph labelling conjectures. Edinah Gnang is the common author, none of the papers seem to be published further, ...
- 1,319
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0
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24
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Minimizing intersections between spanning trees of graph embeddings in polynomial time
Assume I have $N$ complete graphs $G_1, G_2,...,G_N$, and consider their embeddings $E_1, E_2,...,E_N$ in $\mathbb{R}^2$. Is there a (potentially stochastic) polynomial time algorithm to construct ...
- 1
7
votes
2
answers
242
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Prove that $ n \leq d+1 $ under ordering constraints in $\mathbb{R}^d$
Let $x_1, \dotsc, x_n \in \mathbb{R}^d$ and $\theta_1, \dotsc, \theta_n \in \mathbb{R}^d$ be vectors such that for every $k \in [n]$, the following inequality holds:
$$
\langle x_k, \theta_k \rangle &...
1
vote
1
answer
73
views
"Gray code" for $[\omega]^{<\omega}$
Let $\newcommand{\oo}{[\omega]^{<\omega}}\oo$ denote the collection of finite subsets of the set of non-negative integers $\newcommand{\o}{\omega}\o$.
If $A,B$ are any sets, let $A \,\triangle \, B ...
- 48.8k
1
vote
1
answer
98
views
Is there any known upper bound for the local crossing number of a graph drawing in the plane?
The local crossing number ${\rm LCR(G)}$ of a graph $G$ is defined as the least nonnegative integer $k$ such that the graph has a $k$-planar drawing. In other words, it is the smallest possible number ...
- 1,190
4
votes
0
answers
66
views
is a 4-connected planar graph still Hamiltonian after removing an edge?
We know that 4-connected planar graphs are Hamiltonian(by the known Tutte Theorem). Additionally, Thomas and Yu [1] proved that removing two vertices from a 4-connected planar graph still preserves ...
- 1,851
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1
answer
50
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Cycle-Sculpturing with Minimal Vertex-Deletion
given a simple, finite and symmetric graph $G(V,E)$ with $n$ vertices and at least $n$ edges
Question:
how can the smallest set of vertices $W\subset V$ be calculated for which the graph induced by $...
- 13.2k
1
vote
0
answers
51
views
Coarse-graining a hypergraph
$\DeclareMathOperator{\poly}{\mathrm{poly}}$I have asked this question on math.SE here, but couldn't get a satisfactory answer. I have also asked a related question on math overflow here, but haven't ...
- 277
1
vote
1
answer
87
views
Bounds on the number of proper 3-colorings of cubic graphs
Are there known bounds on the number of proper 3-colorings of a 3-regular in terms of vertex count?
- 83
1
vote
0
answers
41
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Asymptotic mixing time and Euclidean probability distance for path graphs
We are given a simple path graph $P(V,E)$ with vertex set $V$ and edge set $E$, having $n=|V|$ nodes. Given an initial distribution $\mathbf{\mu}$ over $V$, let $d_t(\mathbf{\mu},\pi)$ be defined as $\...
- 1,677
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0
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42
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How to determine if two matchings are related by a permutation?
Let $n \geq 2$ be an integer. Let
\begin{align*}
V &= \{(i, j); 1 \leq i, j \leq n \text{ and } i \neq j \} \\
E &= \{ \{v_1, v_2\}; v_1, v_2 \in V \text{ and } v_1 \neq v_2 \}.
\end{align*}
...
- 5,215
0
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0
answers
56
views
Does Forcing conjecture equals to assume the host graph is regular?
Given two graphs $H$ and $G$, the homomorphism density $t(H, G)$ is defined as the proportion of mappings from the vertices of $H$ to the vertices of $G$ that preserve adjacency. Formally,
$$
t(H, ...
- 349
3
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0
answers
92
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Realized graph of majority of permutations
This question was asked several months ago on Math.SE, but remains unsolved.
For any collection of permutations of $\{1,2,\dots,n\}$, we say that it realizes a directed multigraph with $1,2,\dots,n$ ...
- 277
2
votes
0
answers
51
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Subgraphs of random graphs with a given degree sequence
Let $\mathbf{d}=(d_1,\dots, d_n)$ be a given degree sequence with $3\leq d_i\leq \Delta$ for every $i$, where $\Delta$ is constant. Let $G(n,\mathbf{d})$ denote the random graph uniformly distributed ...
- 143
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votes
1
answer
77
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Bipartite matching where every adjacent pair of vertex on the left side of the graph has at least 1 vertex matched
The question is as stated above. I want to devise bipartite matching algorithm where it determies whether every adjacent pair of vertex on the left side of the bipartite graph has at least 1 vertex ...
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0
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45
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Another version of Sidorenko's conjecture(?)
I would like to ask a question about Sidorenko's conjecture. Here is the background of my question:
Quasi-random graphs
A sequence of graphs $(G_n)$ is called quasi-random if it satisfies certain ...
- 349
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0
answers
36
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Construct a maximum matching from a minimum vertex cover in bipartite graph?
Konig's theorem in graph theory says that for a bipartite graph $G$, the size of maximum matching in $G$ is equal to the size of minimum vertex cover of $G$.
Typically, one of the proofs is to ...
- 281
2
votes
0
answers
130
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Does Ising partition function determine the number of $k$-matchings mod $4$ for cubic graphs?
Let $G$ be a cubic graph. It's known that the Tutte polynomial $T_G$ of $G$ on the hyperbola $(x-1)(y-1)=2$ determines the Ising partition function of $G$ and vice versa.
A $k$-matching in a graph $G$ ...
- 9,501
1
vote
1
answer
90
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Characterizing the family of maximal cliques of a cograph
Preamble #1
There are two common equivalent definitions of cographs:
the smallest class that includes $K_1$ and is closed under disjoint union and complementation (or join);
the finite $P_4$-free ...
- 113
2
votes
0
answers
172
views
How many maximal length snakes are there?
This problem was motivated by the classic phone game Snake.
Consider the square grid graph with vertex set $V := \{1, \dots, N\}^2$, for fixed odd positive integer $N$, and an edge between $(x, y)$ ...
- 6,155
5
votes
1
answer
264
views
Approximation of Hamiltonian cycles
Let's define the $\texttt{MinHalfSimpCycle}$ search problem: Given $G=(V, E)$ a complete, undirected graph with weights on the edges. We want a simple cycle in $G$ (each vertex appears in it at most ...
- 53
2
votes
4
answers
212
views
Efficient algorithm for graph problem
Let $D=(V,E)$ be a directed graph, $S,T\subset V$ and $f:V\rightarrow \{1,\ldots, k\}$ a positive, bounded weight-function and $l\in \mathbb{N}$, find a path $v_1,\ldots, v_l\in V$ with $v_1\in S$ and ...
- 157
4
votes
1
answer
141
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Group generated by "adjacent" permutations of graph
Let $G=(V,E)$ be any graph, i.e. $E$ is simply a binary relation over $V$. We say a permutation $\sigma\in \text{Bij}(V)$ of the vertices of $G$ is adjacent if, for all $v\in V$, $(v,\sigma(v))\in E$. ...
- 637
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90
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Definition of Loop in an Oriented Matroid
I had posted this on Stackexchange because I don't believe this is a particlarly difficult question, but there were no answers, so I'm posting it on here now.
I just had a quick question about the ...
- 61
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29
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Explicit Bound in Draganić's Hamiltonicity Result?
Earlier this year, Draganić et al published a remarkable piece of work that resolved Krivelevich and Sudakov's conjecture on the Hamiltonicity of expanders. Here's the abstract:
An n-vertex graph G ...
- 3,979
4
votes
1
answer
252
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What is the resistance between two vertices on the Hanoi-towers graph?
The Hanoi graph $H^n_k$ is the graph with nodes representing states of a Hanoi puzzle with $n$ discs and $k$ pegs, and edges representing the various moves of the discs from peg to peg.
The Hanoi ...
- 2,185
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0
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50
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The uniqueness of a solution Impossible chessboard puzzle
This is a double post.
https://math.stackexchange.com/questions/4981887/the-uniqueness-of-a-solution-impossible-chessboard-puzzle/4981946#4981946
I found a video about a famous puzzle called "...
- 328
2
votes
2
answers
209
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Rank of adjacency matrix of a graph on a sphere all of whose faces have four vertices
Let $G$ be a graph drawn on the sphere such that every face of $G$ has exactly four vertices. Question: can anything be said about the rank of the adjacency matrix of $G$ in terms of other (preferably ...
- 2,964
8
votes
4
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1k
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Counting with trees
Let $\mathcal{U}_n$ denote the set of unrooted unlabelled trees with $n$ edges. For $T\in\mathcal{U}_n$, let $1^{u_1}2^{u_2}\cdots n^{u_n}$ be its degree distribution, that is, $u_i=\#$ of vertices ...
- 43.2k
0
votes
1
answer
123
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Petersen graph does not have a nowhere-zero 4-flow
I try to prove that the Petersen graph does not have a nowhere-zero 4-flow (i.e., over $\mathbb{Z}_4$), but I don't know how a proof could work...
I'm happy about every hint, thank you in advance!
1
vote
1
answer
106
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Iterated optimal transport
Suppose we are interested in two consecutive transport plans (in the Kantorovich formulation). That is, we are given finite sets $X$, $Y$ and $Z$, endowed with probability measures $\mu_X$, $\mu_Y$ ...
- 13
3
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0
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51
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Asymptotic dimension of graph families representing each finite group
Frucht's theorem says every finite group is isomorphic to the automorphism group of a simple graph $G$ (with no loops, multiple edges or directed edges).
There has been interest in finding classes of ...
- 1,926
2
votes
1
answer
100
views
Clique number and a special partition
Let $G=(V,E)$ be a finite, simple, undirected, connected graph, and let $\omega(G)$ denote its clique number. Assume that $G$ has a partition into $m$ independent subsets $U_1,\dots, U_m$ such that ...
- 21
2
votes
1
answer
226
views
Expanders except for commutativity?
What would you call a graph that is an expander except for commutativity, in the following sense?
Say that, from every vertex, you have $d$ edges ($d$ large) labelled $x_1,\dotsc, x_d$. Say that your ...
- 20.2k
2
votes
0
answers
35
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Limiting spectral distribution of a random matrix with specific structure
First, consider an $N \times N$ Hermitian random matrix $V$ from the Gaussian Unitary Ensemble (GUE). It is well known that the empirical spectral distribution of the GUE satisfies the semicircle law ...
- 21
0
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0
answers
25
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Is there a name for a spanner graph that only considers distance to a root node?
A $t$-spanner graph of a set of points $\{p_i\}$ in the plane is a graph $G = (V, E)$ such that for any pair of vertices $p_i, p_j \in V$, the shortest path distance $d_G(p_i, p_j)$ in $G$ is at most $...
- 4,049