Questions about the branch of combinatorics called graph theory (not to be used for questions concerning the graph of a function). This tag can be further specialized via using it in combination with more specialized tags such as extremal-graph-theory, spectral-graph-theory, algebraic-graph-theory, ...

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1
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1answer
92 views

Random graphs with boundary in a game (Tsuro)

Suppose we have an $n \times n$ board and we have $n^2 - 1$ square tiles. These tiles consist of a 8 vertices, two on each edge, and every vertex is connected to precisely one other vertex. These ...
3
votes
1answer
126 views

Graph of graph homomorphisms

For (finite or infinite) undirected, simple graphs $G, H$, let $V_{\text{Hom}} = \{f:G\to H:f\text{ is a graph homomorphism}\}$, and $E_{\text{Hom}} =\big\{\{f,g\}\subseteq V_{\text{Hom}}: ...
3
votes
1answer
132 views

Seymour's second neighborhood conjecture

Does anyone out there know if Seymour's second neighborhood conjecture is still open? if not, I would appreciate any references.
1
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0answers
33 views

Decomposing a weakly chordal graph into disjoint union of co-chordal graphs

A graph G is said to be co-chordal if it is $\bar C_n$-free for any $n \ge 4$. It is weakly chordal if it is $C_n$ and $\bar C_n$ free for all $n\ge 5$. Assume that the induced matching number of $G$ ...
2
votes
0answers
40 views

Approximating a max-cut's intersection with other cuts

(This is a cross-post from the Theoretical Computer Science Stack Exchange.) For the purposes of this question, a cut in a graph $G$ is the edge-set $\delta (S)\subseteq E(G)$ between some vertex-set ...
4
votes
1answer
74 views

Example to show pairwise crossing number is not equal to crossing number

A common point of two edges in a graph drawing that is not an incident vertex is called a crossing. The crossing number $cr(G)$ is defined to be the minimum number of crossings in any drawing of ...
2
votes
0answers
47 views

lower bound on A(k,4,floor(k/2))

A(k,4,r) is the independence number of the Johnson graph J(k,r). What is the best known asymptotic lower bound on A(k,4,floor(k/2)) ? I only obtained ...
6
votes
2answers
151 views

Models for graphs representing real-life networks

I am interested in basic models of graphs (stochastic or deterministic) that are offered for real-life networks (like social networks, the Internet, neuron networks). I will be thankful for answers ...
1
vote
1answer
62 views

Linear algebra formulation for colored node graph isomorphism

(Please see a few paragraphs below by what I mean by “colored node graph isomorphism”.) Some basic definitions for completeness: Given two graphs $G_1=(V_1, E_1)$ and $G_2=(V_2, E_2)$ the graph ...
0
votes
0answers
13 views

Non-discrete modularity measure in graph analysis [closed]

I work in neuroimaging, and right now graph theory is all the rage. Most graph analyses that parcel brain regions into modules do so in a discrete fashion. This might ignore the idea that one brain ...
-1
votes
1answer
30 views

Can Singular Value Decomposition Optimal Newman Modularity? [closed]

I face some problem on the way home. I want to optimize Newman Modularity Q , Can I use SVD to do that? Thanks!
-4
votes
0answers
51 views

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

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?
4
votes
0answers
100 views
+50

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, ...
-3
votes
0answers
46 views

Graph Theory - k-connected graph [closed]

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
vote
1answer
50 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 ...
1
vote
0answers
52 views

Probabilistic proof for expander existence [closed]

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 ...
19
votes
2answers
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 ...
1
vote
1answer
206 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 ...
3
votes
1answer
88 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 ...
5
votes
0answers
93 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
0answers
86 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
0answers
55 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 ...
1
vote
0answers
35 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 ...
4
votes
2answers
148 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 ...
7
votes
1answer
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 ...
25
votes
0answers
537 views

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 = ...
3
votes
2answers
132 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
1answer
115 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 ...
9
votes
3answers
667 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 ...
4
votes
4answers
283 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 ...
2
votes
1answer
39 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
0answers
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 ...
2
votes
1answer
120 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 $|V| = n$, $\Delta(G) \leq \Delta$ ...
3
votes
1answer
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 ...
1
vote
1answer
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 ...
0
votes
1answer
61 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 ...
6
votes
1answer
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 ...
0
votes
1answer
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 $G_S$ ...
0
votes
0answers
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 ...
1
vote
0answers
54 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
3answers
351 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 ...
2
votes
0answers
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?
3
votes
1answer
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 ...
3
votes
1answer
103 views

Infinite non-splittable graphs

Let $G=(V,E)$ be a graph. For $v\in V$ we set $N(v)=\{w\in V:\{v,w\}\in E\}$. We say that $G$ is splittable if there are $S,T\subseteq V$ with $S\cap T=\emptyset$ and $S\cup T = V$ such that for all ...
1
vote
2answers
123 views

A certain matrix associated to graphs

I am not very familiar with graph theory, but I need some results for my work. Thus, the question is, whether the following has already been studied and where I can find it. Let $G=(V,E)$ be an graph ...
5
votes
0answers
247 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 ...
-2
votes
1answer
108 views

About structure of the set of perfect matchings of $K_{n,n}$

Are there any special properties known about the set of perfect matchings of $K_{n,n}$? Like any global structure of this set? Some natural way to partition it? Like is there some algebraic structure ...
8
votes
0answers
151 views

Lovasz's Path removal conjecture

The Lovász Path Removal Conjecture states: For any positive integer $k$, there exists a minimum positive integer $f(k)$ such that, for any two vertices $x$, $y$ in any $f(k)$-vertex-connected ...
7
votes
0answers
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 $g_{12}=g_{21}=g_{13}=g_{31}=1$ ...
6
votes
1answer
177 views

Isometries of some simple Cayley graphs

Consider a Cayley graph of a group $G$ with respect to a symmetric finite generating set $S$. There are some obvious candidates to isometries of this graph - for example, translation by elements of ...