# Tagged Questions

**4**

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100 views

### Prove or disprove a claim about covering a polytope by convex polytopes in a certain way

Here is the claim:
Given a polytope $K$ in a unit ball in $\mathbb{R}^d$, there exists a universal constant $C(d)>0$ depending only on $d$ and a countable collection of convex polytopes $\{K_i\}_{...

**4**

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127 views

### Application of finding shortest paths on Cayley graphs

For a fixed integer number $m$, Consider Cayley graph defined by all m-cycles in Symmetric group $Sym(n)$.
I know that for $m=2$,
there are some applications of finding shortest paths (or distance ...

**4**

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45 views

### Digraph weak connectivity in $O(V)$ space and $O(V+E)$ time

A digraph is called weakly connected if its underlying undirected graph is connected.
You are given a digraph $G$ with $V$ vertices and $E$ edges as a read-only data structure consisting of lists of ...

**4**

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99 views

### Counting vertex-permutations of a finite tree which rip all edges

Given a finite tree $T$ with $n$ vertices labelled $1,\dots,n$, we say that a permutation $\sigma$ of $1,\dots,n$ rips all edges if $\{\sigma(i),\sigma(j)\}$
is never an edge for every edge $\{i,j\}$ ...

**4**

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176 views

### Connection between connectivity and cohesion of a graph

Tutte [1] proved that, for every $3$-connected graph $G$ and vertices $u$ and $v$, there exists a nonseparating $uv$-path.
A graph $G$ is $t$-cohesive if $G$ is connected, has at least two vertices, ...

**4**

votes

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154 views

### What is known about the complexity of this covering problem?

Let $G=(V,E)$ be a graph. A vertex set $X\subseteq V$ is called critical if $X\neq\emptyset$ and no vertex in $V\setminus X$ is adjacent to exactly one vertex in $X$. The problem is to find a vertex ...

**4**

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126 views

### $\kappa$-impediments (according to Shelah, Nash-Williams, Aharoni)

Let $\Gamma = (M, W, K)$ be a bipartite graph, that is $M, W$ are sets and $K\subseteq M\times W$. If there is an injective function $f:M\to W$ such that $f\subseteq K$ we say $f$ is an espousal and $\...

**4**

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178 views

### Hypercube edge-coloring problem

Question: Is there a pairing (a fixed point free involution) of the vertices of the $n$-dimensional cube graph, and a $2$-coloring of its edges such that the number of color changes needed to get from ...

**4**

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177 views

### Sets of spreads in graphs

Let $G$ be a graph. A $k$-spread is a set of cliques of order $k$ which partition the vertex set (so $k|n$, where $n$ is the number of vertices).
A partial $k$-resolution of $G$ is a set of pairwise ...

**4**

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203 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 ...

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97 views

### Actions of amenable groups on graphs with uncountably many ends

Let $G$ be a finitely generated amenable group acting transitively on an amenable Schreier graph $S$. Is it possible for $S$ to have uncountably many ends? An amenable graph with uncountably many ends ...

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182 views

### Counting Problems where Labeled is Known but Unlabeled is Not

Cayley's formula states that the number of labeled trees on $n$ vertices is $n^{n-2}$. There are many nice proofs of this compact formula.
To contrast, counting unlabeled trees is considerably harder....

**4**

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140 views

### What do we know about isospectral Cayley graphs?

Let $\Gamma_1=\Gamma_1(G_1,S_1)$ (respectively $\Gamma_2=\Gamma_2(G_2,S_2)$) be a Cayley graph for the group $G_1$ and the finite generating set $S_1$ (respectively $G_2$ and the finite generating set ...

**4**

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120 views

### More 3-connected cubic graphs with all 2-factors of same cycle type?

The setup is as in this question: Let $G$ be a 3-connected cubic graph. If all 2-factors of $G$ are isomorphic (as graphs), i.e. all have the same partition $\pi$ as cycle type, we'll say that $G$ is ...

**4**

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72 views

### What is the probability of interpolating the Tutte polynomial of a planar graph from the values at the two hyperbolas?

The Tutte polynomial
is a bivariate polynomial with positive integer coefficient which is a graph
invariant and can be defined recursively.
Evaluating it is $\#P$-complete even when restricted to (...

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130 views

### Independence Number of K4-free planar graphs

The independence ratio is defined as the size of the maximum independent set divided by $n$. By the 4-color theorem, every planar graph with $n$ vertices has independence ratio at least $1/4$. Clearly,...

**4**

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158 views

### Does the concept of connective constant make sense for any tiling of the plane?

First let me define what is the "connective constant" of a two dimensional lattice.
Let $c_{n}$ denote the number of $n$ step self-avoiding walks starting from a fixed origin point in the lattice. ...

**4**

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152 views

### Genus of the graph $K_{m,2,2,2}$

What is the genus of this complete $4-$partite graph, $K_{m,2,2,2}$, where $m \in \mathbb{N}$?
Thanks in advance.

**4**

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297 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 is ...

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537 views

### Does this graph property have a name?

I'm interested in a family of properties of connected simple graphs that comes up in percolation theory.
Let $G$ be a simple connected graph. Now consider the set of subgraphs of $G$ that I will call ...

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92 views

### Upper bound on size of obstruction set for wye-delta-wye reducible graphs

A graph is $Y \Delta Y$-reducible if it can be reduced to an empty graph by the following operations:
$Y \leftrightarrow\Delta$ transforms;
Replacing multiple edges with single edges (parallel ...

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126 views

### What are the best bounds to date on the maximum girth of a cubic graph?

The girth of a graph is the length of its smallest cycle. Studying the maximum possible girth for a $k$-regular graph on $n$ vertices is a very well-studied problem.
In the 1988 paper "Ramanujan ...

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569 views

### Questions about dessin d'enfants, trees and their Shabat polynomials

This will be a series of questions, a few of which have been troubling me for quite a while now. Before I jump right in, let me first introduce a few notions which I will assume.
(Note: All of these ...

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179 views

### Rough structure of the double coset space/Graph bijections up to automorphisms

I am dealing with bijective maps $\pi:\Gamma_1\to \Gamma_2$ between two graphs with the same number of vertices $N=O(10)$.
The graphs have a significant automorphism group (these are disconnected ...

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163 views

### Equivalent paths in graphs

Let $G$ be a finite, planar graph. In this question I consider a path in $G$ to be a finite sequence of vertices $v_1,\dots,v_n$ of $G$, where $v_i$ is adjacent to $v_{i-1}$ for $i=2,\dots,n$. A ...

**4**

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101 views

### Metrized categories

Motivation: Let $\Gamma = (V,E)$ be a directed graph. To each edge $e \in E$, choose a value $\kappa^e \in \mathbb R$, representing the cost of transporting one unit of "stuff" through the edge. Let $\...

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197 views

### Graph distance of close points within the minimum spanning tree

My question is the following: Given $N$ uniform IID points $X=(X_1,...,X_N)$ on the unit cube of $\mathbb{R}^d$, take $X_1$ and another point, say $X_{(1)}$, "close" to $X_1$ (i.e. connected to $X_1$ ...

**4**

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141 views

### Graph drawing maximizing the volume of the convex hull

Given a graph $G=(V,E)$ and a length function $\ell:E\to\mathbb{R}_+$.
An embedding of the graph into the $d$-dimensional Euclidean space is a map $f:V\to\mathbb{R}^d$ such that $||f(u)-f(v)||=\ell(uv)...

**4**

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250 views

### origin of the notion of “network” in graph theory

In current graph theory, a "network" is a precisely defined object: It is a directed graph associated with a function, the "capacity", which is defined on the edge set and has certain specific ...

**4**

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573 views

### determinant of fibonacci-sum graphs

We have a simple graph with vertices $\{v_1, v_2, ... v_n\}$.
The adjacency matrix of this graph is $A= (a_{ij})$ so that
$a_{ij}=1$ if $i+j$ belongs to the Fibonacci sequence;
$a_{ij}=0$ ...

**4**

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376 views

### Tensor product of quivers

As a special case of a general construction I have constructed "accidentally" a tensor product of quivers aka directed multigraphs (aka directed graphs for category theorists). Probably this is ...

**4**

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176 views

### Shannon capacity of all graphs of order 6

In Shannon's paper, "The Zero Error Capacity of a Noisy Channel", he says that the Shannon capacity of all graphs of order 6 are determined, except for four exceptions. That is, for all but the four ...

**4**

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429 views

### System of Equations Upper Bound

I asked a related question on math.stackexchange here but would now like to obtain a better bound. This question comes from a graph theory problem. I'll restate the new question here:
For $i=1,2,\...

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372 views

### Expected number of components with multiple cycles in a subgraph of a square lattice

Short version
Is there an understanding of the emergence and subsequent disappearance of components with zero, one, or more cycles in a random subgraph of a square or cubic lattice, as the edge-...

**4**

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**0**answers

163 views

### Best lower bound for proof complexity of graph asymmetry

Graph automorphism problem ( GA) of determining whether a graph has a nontrivial automorphism is a good candidate for a problem in $NP$-intermediate. I'm looking for references that study the ...

**4**

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**0**answers

165 views

### Bounds on numbers of matchings of given sizes in bipartite graphs

I am interested in the following question:
For which sets ${m_1,\ldots m_k}$ of positive integers do there exist bipartite graphs having exactly $m_i$ matchings of size $i$ for each $1\leq i \leq k$, ...

**4**

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**0**answers

428 views

### Smallest matrix covered by many random n by n matrices

We say that a matrix $M$ can be covered by a (smaller) matrix $N$ if every entry in $M$ is contained in some submatrix of $M$ that exactly equals to $N$, up to reordering the rows and columns of $N$. ...

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379 views

### SWAT vs Rioters (cops vs robbers variant)

I thought of this while at the Combinatorial Potlatch at Seattle University, where Peter Winkler gave an excellent talk on Cops vs Drunken Robbers. I'll just open it up to the floor. The problem ...

**4**

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**0**answers

218 views

### Applying the amplification trick + probabilistic method on connected graphs

First of all, I apologise if my question is not very specific. I am not really looking for a concrete answer but rather some hints and pointers what could be used/applied in this situation. A concrete ...

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417 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 ...

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545 views

### edge disjoint covers of graphs by paths

Let $G=(V,E)$ be a graph where every "internal" vertex has degree 4, and every "external" vertex has degree $\le 3$. What can be said about this graph if it can be covered by a collection of edge-...

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417 views

### Generating random polygons from a given triangulation of points

Given a triangulation $T$ of a planar set point $S$, we would like to randomly generate a polygon (hamiltonian cycle) $P$.
However, it has been proved that Hamiltonian Circuit Problem on maximal ...

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176 views

### Complex Laplacians in spectral decompostions of graphs

The Laplacian of a graph is a useful tool in many kinds of graph decomposition problems. Since inspection of the eigenvector corresponding to the second smallest eigenvalue (the Fiedler vector) yields ...

**4**

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328 views

### Combinatorics of signed oriented graphs/skew-symmetric matrices

Consider a "complete" signed graph on $n$ vertices indexed by
$1,2,\dots,n$, that is, a graph in which any two distinct vertices $i$ and $j$ are connected by an oriented edge.
For each pair of ...

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votes

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112 views

### Limit Distribution of Topological Form of Polyhedra with Large Number of Edges

Consider the set of all topologically inequivalent polyhedral graphs with $k$ edges, the number of which is given by Sloan sequence A002840 (1,0,1,2,2,4,12,22,58,158,448)).
Now define a topological ...

**4**

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461 views

### Natural models of graphs?

Motivation
I want to capture the notion of natural models of finite graphs: How can natural predicates and natural relations on a given natural base class $D$ be defined? If this succeeds the ...

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65 views

### Is the chromatic number of the graph of the permutahedron known?

The permutahedron $\Pi_n$ is the polytope that is the convex hull of all permutations of the vector $(1,2,...,n)$. There are many results on its structure, but I couldn't find a result on the ...

**3**

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32 views

### how to study the size of basins of attraction on a graph

I have a certain finite (but huge and without an apparent pattern, so that only numerical studies seem feasible) graph $G = (V,E)$, and a function $f: V \rightarrow \mathbb{R}$. On each edge $e = (u,v)...

**3**

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49 views

### Hypergraph edge colouring

I'm interested in knowing if finding the edge-chromatic number of a $k$-uniform $k$-partite hypergraph is NP-hard for $k\geq 3$ Could anyone provide a reference for the result? By edge-chromatic ...

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85 views

### What is the complexity of finding a third Hamilton Cycle in cubic graph?

According to Smith Theorem: if a cubic graph has a hamilton circuit then it must have a second one. SMITH : Given a Hamilton circuit in a 3-regular graph, find a second Hamilton circuit. It is known ...