# Tagged Questions

**17**

votes

**1**answer

813 views

### A Question on 1, 2 ,3 Conjecture

The 1, 2, 3 conjecture is well-known:
If $G$ is a simple graph which is not $K_2$ then one can assign a number among $1, 2, 3$ to every edge such that if we label each vertex with the sum of the ...

**-4**

votes

**0**answers

39 views

### how should one locate ambulance stations so as to best serve the needs of the community..tnx [on hold]

how should one locate ambulance stations so as to best serve the needs of the community
i don't know what algorithm to use, any suggestion/s?

**3**

votes

**0**answers

16 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, ...

**1**

vote

**1**answer

283 views

### Random walk on the hypercube

Consider the hypercube $Q_4$. I would like to know how to compute the number of steps of a random walk in this graph such that the probability to be at a vertex is a given number $x$. I think I just ...

**-3**

votes

**0**answers

41 views

### Graph Theory - k-connected graph [on hold]

I am trying to understand the concept of k-connected graphs in graph thoery. Reference books state that a graph G is k-connected if G is connected and if its vertex connectivity is greater than or ...

**-1**

votes

**0**answers

152 views

### Limit Group decomposition

I would need a clarification about a statement in the article Limit groups and groups acting freely on $\mathbb{R}^n$-trees by Vincent Guirardel.
First recall that a limit group is a finitely ...

**1**

vote

**1**answer

40 views

### Expected number of leaf nodes in some theoretical graph models

If a leaf node of a graph refers to a node having the degree of 1, how can one compute the expected number of leaf nodes of:
(A) a random graph (e.g., Erdos-Renyi graph),
(B) a small-world graph ...

**4**

votes

**1**answer

285 views

### Generating non-isomorphic graphs by adding edges to a given graph

This question is in a way related to the one I posted on math.se. Since the question there did not produce any final answer I am trying my luck here!
I am given a fairly large graph $G$ and subsets ...

**1**

vote

**1**answer

2k views

### 3D Delaunay Triangulation -> Euclidean Minimum Spanning Tree

I read that the Euclidean Minimum Spanning Tree (EMST) of a set of points is a subgraph of any Delaunay triangulation. Apparently the easiest/fastest way to obtain the EMST is to find the Deluanay ...

**20**

votes

**17**answers

24k views

### Good programs for drawing graphs ( directed weighted graphs )

Does anyone know of a good program for drawing directed weighted graphs?

**1**

vote

**0**answers

46 views

### Probabilistic proof for expander existence [on hold]

I am new to probabilistic proofs and trying to understand them better. Apparently, a common probabilistic proof focuses on the existence of expanders (eg. vertex expanders).
I've been using the search ...

**18**

votes

**2**answers

1k views

### An unfair marriage lemma

I am looking for a citeable reference to the following generalization of Hall's Marriage Theorem:
Given a bipartite graph of boys and girls. In addition to gender difference, they are divided into ...

**11**

votes

**2**answers

579 views

### Gale-Shapley stable marriage theorem: can we entrust matchmaking to monkeys?

Disclaimer: This is a question I have not done any real research about. I asked it myself some 5 years ago, and back then I had no idea where to start. Now I have some texts on stable matchings lying ...

**0**

votes

**1**answer

171 views

### Efficient isomorphic subgraph matching with similarity scores

I'm a computer vision PhD student, and I'm looking for an efficient approximation to the following problem, which could end up helping in image to image matching. Failing that, pointers to relevant ...

**3**

votes

**1**answer

80 views

### Create matrix containing values in [0,1] where sum of all diagonals and anti-diagonals is fixed

The problem I am facing sounds at first glance pretty simple. However, as very often, it seems more complicated than I first assumed:
I want to calculate a matrix $P = (p_{j,k}) \in \mathbb{R}^{n ...

**2**

votes

**0**answers

54 views

### Edge-disjoint path-systems in infinite digraphs

Let $ D=(V,A) $ be a directed graph without backward-infinite paths and let $ \{ s_i \}_{i<\lambda},U \subset V $ where $
\lambda $ is some cardinal. Assume that for all $ u\in U $ there is a ...

**-4**

votes

**0**answers

62 views

### Graph theory and topology [on hold]

I have related topological ideals with vertex magic totallabeling in graph theory. Is there any possibility to relate vertex magic totallabeling with generalized topology in a very interesting way? ...

**4**

votes

**0**answers

85 views

### A digraph related to permutations

A finite sequence of distinct real numbers of length $n$ determines a linear order of $\{1,\ldots,n\}$, by mapping position to rank; call this the permutation of the sequence.
Consider the following ...

**3**

votes

**0**answers

82 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 ...

**2**

votes

**1**answer

119 views

### Coloring algorithm maximising color difference between neighbors

Consider a graph and a set of ordered colors ${\cal C} = \{1,2,\cdots,C\}$. I want to color each node $i$ with a color $c_i\in{\cal C}$ so as to maximize the minimum color difference between two ...

**24**

votes

**0**answers

477 views

+250

### 3-colorings of the unit distance graph of $\Bbb R^3$

Let $\Gamma$ be the unit distance graph of $\Bbb R^3$: points $(x,y)$ form an edge if $|x,y|=1$.
Let $(A,B,C,D)$ be a unit side rhombus in the plane, with a transcendental diagonal, e.g. $A = ...

**0**

votes

**1**answer

66 views

### Counting the orderings of outward-directed trees where the degree of each vertex is $2$

Let $T$ be a connected directed tree with the following properties:
The degree of each vertex of $T$ is at most $2$ (I am sure there is a name for such a graph but I do not know it).
$T$ has a ...

**5**

votes

**0**answers

244 views

### Have topographs been studied before?

This is my first post on MO so I hope this question is suitable. I have quite a few definitions which I will need to state before my questions at the end of this post. Please let me know if anything ...

**1**

vote

**0**answers

34 views

### Mappings between adaptive networks and Markov processes

Are there any known mappings between adaptive networks models (i.e. graph model representations of networks where the internal vertex dynamics and connectivity topology can change subject to specific ...

**5**

votes

**1**answer

305 views

### How to find or constrain “particularly good” (two-sided) spectral expanders?

I'm new to graph theory, but a response to a question I asked a while ago introduced me to the concept of expander graphs.
A k-regular graph (henceforth "graph") on n nodes has eigenvalues k = λ1 ≥ ...

**-1**

votes

**0**answers

46 views

### Looking for an example of a contour integral with matrix entries [closed]

Let $A$ be a matrix (if needed assume it to be the adjacency matrix of graph). Let one be given two functions $P(z)$ and $Q(z,A)$ such that both are polynomials in $z$ and $A$, where $z$ is some ...

**7**

votes

**1**answer

112 views

### Is there a Degenerate Dependency Local Lemma?

The Lovasz Local Lemma has several generalizations, with names usually starting with L, such as Lopsided or Lefthanded.
Here I ask whether another possible generalization (for which I could not yet ...

**1**

vote

**1**answer

106 views

### Hamming graph and independent sets

I'm defining the Hamming graph $H(d,q)$ in the usual way, so we have a set $S$ of $q$ elements, the hamming graph $H(d,q)$ has vertex set $S^{d}$ (the set of all ordered $d$-tuples of elements of $S$) ...

**4**

votes

**2**answers

134 views

### How many simple cycles can a graph with $n$ vertices and $m$ edges have?

I am mainly interested in the smallest number of simple cycles a graph with $n$ vertices and $m$ edges must have.
For example, if $m\le n-1$, this number is $0$, then if $n\le m \le 3(n-1)/2$, it is ...

**0**

votes

**1**answer

64 views

### Petersen 2-factor decomposition theorem for directed graphs

Petersen proved that every 2k-regular graph can be decomposed into k disjoint 2-factors. I would like to know that is it true that if G is a directed regular graph (d_out(v)=d_in(v)=k), then can G be ...

**12**

votes

**1**answer

764 views

### A generalization of the triangle counting problem for simple weighted graphs

One nice identity is $$tr(A^3)/6$$ which counts the number of triangles of a graph represented with adjacency matrix $A.$ It also implies that triangle counting can be performed in subcubic time.
...

**9**

votes

**3**answers

658 views

### Is every graph an edge-crossing graph?

Consider a circular drawing of a simple (in particular, loopless) graph $G$ in which edges are drawn as straight lines inside the circle. The crossing graph for such a drawing is the simple graph ...

**3**

votes

**2**answers

131 views

### The Hadwiger number of $L(K_n)$

For $n\in\mathbb{N}$ we consider the set $\{1,\ldots,n\}$ and define the line graph $L(K_n)$ of the complete graph $K_n$ as follows:
$V(L(K_n)) = \big\{\{a,b\}: a,b\in \{1,\ldots, n\}, a\neq b ...

**1**

vote

**1**answer

113 views

### Implication between Erdös-Faber-Lovasz conjecture and Hadwiger's conjecture?

The Erdös-Faber-Lovasz conjecture and Hadwiger's conjecture can be stated in a very similar form:
Erdös-Faber-Lovasz conjecture: for all finite simple undirected graphs $G=(V,E)$ we have $\chi(G)\leq ...

**4**

votes

**4**answers

268 views

### Request for examples of 4-regular, non-planar, girth at least 5 graphs

Edit: As David Eppstein points out (in his answer below) the assumption that the graph is non-planar is redundant.
Thank you to everyone who answered/commented.
I have a problem about geometric ...

**5**

votes

**1**answer

305 views

### Cospectrality and dimension of graphs

Firstly, I apologize if the question is long. I appreciate any helpful answers and ideas.
In the following all graphs are simple and connected.
Let $G$ be graph with vertex set ...

**2**

votes

**1**answer

38 views

### Adding vertex-disjoint edges to reduce the diameter

Let $G=(V,E)$ be a finited connected graph, $V\neq \emptyset$. Let $[V]^2 := \big\{ \{v,w\}: v, w \in V\text{ and } v\neq w\big\}$. Given $F\subseteq [V]^2$ we say that $F$ is a vertex-disjoint ...

**1**

vote

**0**answers

9 views

### Is it true that centrality measures in SNA are indicative for most important vertices only?

I read about the limitations of centrality measures on Wikipedia. It says that centrality measures are good only for identifying top most important nodes in a social network. Their relative values can ...

**1**

vote

**1**answer

114 views

### Is the number of vertices bounded for fixed max degree and fixed diameter?

Are there positive integers $\Delta, d$ such that the following statement is true?
For every $n\in \mathbb{N}$ there is a graph $G = (V,E)$ such that $|V| = n$,
$\Delta(G) \leq \Delta$ ...

**1**

vote

**1**answer

33 views

### Connected components of a sum-graph over an infinite set

Given $S\subseteq \mathbb{N}$, we associate to $S$ the sum-graph $G_S = (\mathbb{N}, E)$ where $$E = \big\{\{m,n\}: m,n \in \mathbb{N} \text{ and } m+n\in S\big\}.$$
Is there an infinite subset ...

**6**

votes

**1**answer

111 views

### Diameter of sum-graph over a non-meager set

Given $S\subseteq \mathbb{N}$, we associate to $S$ the sum-graph $G_S = (\mathbb{N}, E)$ where $$E = \big\{\{m,n\}: m,n \in \mathbb{N} \text{ and } m+n\in S\big\}.$$
We say that $S\subseteq ...

**3**

votes

**1**answer

91 views

### Diameter of sum-graph over a meager set

We say that $S\subseteq \mathbb{N}$ is meager if $$\text{lim sup}\frac{S\cap\{1,\ldots, n\}}{n} = 0.$$
Given $S\subseteq \mathbb{N}$, we associate to $S$ the sum-graph $G_S = (\mathbb{N}, E)$ where ...

**0**

votes

**1**answer

60 views

### Edge-disjoint cycles in graphs

Given a graph $G=(V,E)$ and a fixed integer $k$ are there any algorithms known which would find the maximum number of edge-disjoint cycles of length $k$ in $G$? If not is there a proof that this ...

**0**

votes

**1**answer

39 views

### Sum-graph over an infinite set

Given $S\subseteq \mathbb{N}$, we associate to $S$ the sum-graph $G_S = (\mathbb{N}, E)$ where $$E = \big\{\{m,n\}: m,n \in \mathbb{N} \text{ and } m+n\in S\big\}.$$
If $S$ is infinite, is $G_S$ ...

**2**

votes

**0**answers

37 views

### Is the size of maximum matching in vertex transitive 3-uniform hyper-graph on $n$ vertices always $\Omega(n)$?

What is the best known lower bound on the size of the maximum matching in a vertex transitive $3$-uniform hyper-graph?

**0**

votes

**0**answers

40 views

### Quasi-transitive decomposition of a transitive graph

Let $G=(V,E)$ be a simple digraph that is semi-complete (ie. there's at least one arc between each unordered pair of vertices) and quasi-transitive (ie. its complement is transitive).
Is it true that ...

**7**

votes

**0**answers

266 views

### Does this inequality always hold?

Denote the adjacency matrix of a given undirected graph by $g$. It is an $n$-by-$n$ symmetric Boolean matrix with elements on the diagonal to be zero ($n\geq 3$). Let $g_{12}=g_{21}=g_{13}=g_{31}=1$ ...

**1**

vote

**0**answers

52 views

### What's the complexity of the one sink directed subgraph isomorphism problem?

I am considering trying a new approach for the subgraph isomorphism problem in my PhD, but it just seems to work well for digraphs of one sink. By working well I mean some promise of not having to ...

**10**

votes

**3**answers

347 views

### The diameter of a certain graph on the positive integers

Let $G(n)$ be the graph whose vertices are the positive integers $1,2,3,4, \ldots, n$ two of which are joined by an edge if their sum is a square. Is the diameter of this graph 4 for all sufficiently ...

**3**

votes

**1**answer

64 views

### Dense high-degree sub-graphs of dense graphs

Let $G$ be a graph with $n$ vertices and $m$ edges, and let $d=\lfloor\frac{m}{n}\rfloor$ be the rounded-down average-degree. A lemma that is attributed to Erdos says that $G$ has a non-empty induced ...