**6**

votes

**3**answers

150 views

### Borel coloring of a graph on the set of all functions $f:\mathbb{N}\to\mathbb{N}$

The following question was asked in a comment by Joel David Hamkins in Graph on the set of all functions $f:\mathbb{N}\to\mathbb{N}$.
Let $V$ be the set of all functions $f:\mathbb{N}\to\mathbb{N}$. ...

**-1**

votes

**1**answer

40 views

### How does deletion-contraction affect chromatic number? Can it increase chromatic number?

Question: In graph theory, contracting an edge or deleting an edge are basic operations in many topics such as graph minors or Wagner's theorem on planar graphs. And I'm interested in how these ...

**1**

vote

**0**answers

13 views

### Minimality of maximal expansions of a hypergraph cover

This is a follow-up question to Maximal expansions of strongly minimal covers of hypergraphs -- for definitions refer to that question.
Does every strongly minimal cover have a maximal expansion that ...

**2**

votes

**1**answer

11 views

### Maximal expansions of strongly minimal covers of hypergraphs

Let $H = (V,E)$ be a hypergraph, that is $V$ is a set and $E \subseteq {\cal P}(V)$. We assume $\bigcup E = V$. Moreover we assume that every $e\in E$ is contained in some maximal member $e'\in E$ ...

**3**

votes

**1**answer

131 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 ...

**2**

votes

**1**answer

155 views

### Graph on the set of all functions $f:\mathbb{N}\to\mathbb{N}$

Let $V$ be the set of all functions $f:\mathbb{N}\to\mathbb{N}$. Let $$E:=\big\{\{f,g\}: f, g \in V \land f\neq g\land \exists k\in \mathbb{N} \text{ }\forall n\in\mathbb{N}\setminus\{k\} (f(n) = ...

**0**

votes

**1**answer

78 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 ...

**10**

votes

**1**answer

221 views

### Graph $G$ with $\omega(G) = 2$ but $\chi(G) \geq \aleph_0$

Given an infinite cardinal $\kappa$, is there a graph $G$ that has no clique consisting of more than 2 points, but $\chi(G) = \kappa$?

**8**

votes

**1**answer

683 views

### When the Lovász theta-function saturates its upper bound

The Lovász $\vartheta$-function of a graph $G$, $\vartheta(G)$, is well-known to be "sandwiched" between the independence number of the graph, $\alpha(G)$, and the chromatic number of its complement, ...

**7**

votes

**1**answer

1k views

### Cycling through the Zeta Garden: Zeta functions for graphs, cycle index polynomials, and determinants

Zeta functions abound in mathematics. Audrey Terras describes in Zeta Functions and Chaos three zeta functions--the zeta fct. of a projective non-singular algebraic variety; the Artin-Mazur zeta ...

**5**

votes

**0**answers

37 views

### Perfect matchings of a regular, uniform, partite hypergraph

This is in relation to the question here. What, if any, are the known conditions for the existence of a perfect matching for a $r$-regular, $r$-uniform, $r$-partite hypergraph. I specifically ...

**1**

vote

**1**answer

80 views

### Does every connected vertex transitive graph on $n$ vertices (except for $C_n$) have minimum feedback vertex set of size $\Omega(n)$?

Feedback vertex set is a set of vertices whose removal leaves an acyclic graph.
It is known that every vertex transitive graph on $n$ vertices has minimum vertex cover of size $\Omega(n)$. It is also ...

**6**

votes

**1**answer

323 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**

vote

**1**answer

193 views

### Threshold for perfect Matchings in Bipartite graph

Consider the random bipartite graph with vertex classes of size $n$ and each edge being present independently with probability $p(n)$.
I know one way to prove the threshold of a perfect matching is ...

**-2**

votes

**0**answers

53 views

### Directed Graph Equivalence Class

Consider the following conversion involving directed graphs.
To convert from $\mathcal{G}$ to $\mathcal{G}^u$ (they have the same number of vertices):
$V_i \rightarrow V_j$ in $\mathcal{G}^u$ iff ...

**0**

votes

**1**answer

73 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 ...

**1**

vote

**2**answers

256 views

### Expected matching in a bipartite graph

Consider a random bipartite graph constructed on vertex classes of size $n$ with each edge present independently with probability $p$. How could I go about calculating the size of the expected ...

**26**

votes

**16**answers

7k views

### Linear Algebra Proofs in Combinatorics?

Simple linear algebra methods are a surprisingly powerful tool to prove combinatorial results. Some examples of combinatorial theorems with linear algebra proofs are the (weak) perfect graph theorem, ...

**4**

votes

**1**answer

269 views

### Generalizing the circle packing theorem to 3-dimensions

The circle packing theorem (Koebe–Andreev–Thurston theorem) states that every finite planar graph is the nerve of some disk packing in the plane, where the nerve of a packing $P$ is a graph $G=(V,E)$, ...

**12**

votes

**1**answer

801 views

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

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

**-5**

votes

**0**answers

28 views

### How to do I do a one to many bipartite matching of two sets? [closed]

I have a set $C$ and another set $A$. Both the set have integers such that any element in $C$ is greater than every element in $A$. I need form a one to many matching between set $C$ and $A$ such that ...

**5**

votes

**1**answer

124 views

### Regular epimorphisms in the category of simple undirected graphs

Let $\textbf{Grph}$ be the category whose objects are graphs $G = (V,E)$ such that $V$ is a set and $E \subseteq \mathcal{P}_2(V) := \{\{a,b\} \subseteq V: a\neq b\}$. We sometimes write $E(G)$ for ...

**4**

votes

**1**answer

92 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 ...

**7**

votes

**1**answer

256 views

### Under $\neg CH$, have countable unions of rationally independent numbers inner measure zero?

In their 1943 paper On non-denumerable graphs, Erdos and Kakutani suggest as likely the following proposition.
(EK*) Suppose CH fails and $\lbrace M_n : n \in \omega \rbrace$ is a countable family of ...

**1**

vote

**1**answer

164 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 ...

**22**

votes

**1**answer

1k views

### Algebraic proof of Five-Color Theorem using chromatic polynomials by Birkhoff and Lewis in 1946

I'm guessing everyone is familiar with Four Color Theorem which was proved by Appel and Haken using computers. A weaker version of this theorem is Five Color Theorem which states that a planar graph ...

**5**

votes

**2**answers

183 views

### Condition(s) for the full autormophism group $\operatorname{Aut}(C(G, S))$ of the Cayley graph of $G$ to be isomorphic to $G$

If $\Gamma = C(G, S)$ is the (undirected) Cayley graph of a finite group $G$ with generating set $S$, then $G \le \operatorname{Aut}(\Gamma)$, the "full" automorphism group of $\Gamma$.
When is ...

**9**

votes

**1**answer

182 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 ...

**2**

votes

**0**answers

20 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 ...

**5**

votes

**0**answers

60 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 ...

**5**

votes

**1**answer

236 views

### A claim from “Graph minors - a survey” by Robertson and Seymour

Can someone give me a proof sketch for this:
Let $\mathscr{P}_n$ be the set of all graphs which do not contain a path on $n$ vertices as a subgraph. Define the type of a graph inductively as: the type ...

**6**

votes

**1**answer

323 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 ...

**1**

vote

**1**answer

1k views

### Does this linear algebra construction based on a graph have a name, and where has it been studied?

In the paper Kochen-Specker set with seven contexts by Lisonek, Badziag, Portillo and Cabello, the following construction is used :
Question : Have such constructions been used elsewhere, and if so ...

**0**

votes

**1**answer

54 views

### Connectedness of the complements of the connected subsets

EDIT: My original foolish version was instantly destroyed by Dylan Thurston; it consisted of questions 1 & 2 below. Thus now only new question 0 remains to be answered.
Let $\ X:=M^n\ $ be a ...

**2**

votes

**1**answer

106 views

### Probability of relations in network

Imagine, i have a predicate $\text{friends}(x_1, x_2)$ and I know that $p(\text{friends}(x_1, x_2)) = p_2$. If I generate a world of $n$ people ($x_1$ to $x_n$), I expect there to be $\binom{n}{2}p_2$ ...

**1**

vote

**0**answers

80 views

### Are back edges mandatory in Ford Fulkerson algorithm?

Consider the algorithm of Ford Fulkerson where, for each iteration, you add flow along a path equal to the maximum residual capacity along this path.
Does it exist, for every network, a choice of ...

**4**

votes

**2**answers

165 views

### Vertex expansion of the Hamming graph

Let $G$ be a graph, and for every $S \subseteq V$, let $N(S)$ denote the neighborhood of $S$. The vertex expansion of $G$ is
$$ \min_{S\subseteq V, |S|\le |V|/2} \left\{ \frac{|N(S)|}{|S|} ...

**3**

votes

**1**answer

77 views

### Average degree of neighbors in a simple graph (-> Friendship paradox)

Given a simple, undirected graph $G=(V,E)$ and $v\in V$ we set $N(v) = \{w\in V:\{v,w\} \in E\}$ and $\text{deg}(v) = |N(v)|$.
The average degree of the neighbors of a vertex $v$, or $\text{ad}(v)$, ...

**9**

votes

**5**answers

350 views

### Do there exist sparse graphs with large crossing number?

Does there exist a sequence of graphs $\{ G_n \}$ such that
$G_n$ has $n$ vertices,
the number of edges of $G_n$ is $O(n)$, and
the crossing number of $G_n$ is $\Omega(n)$?
In particular, do ...

**2**

votes

**1**answer

104 views

### Regular and extremal monomorphisms in the category of graphs

Let $\textbf{Grph}$ be the category whose objects are graphs $G = (V,E)$ such that $V$ is a set and $E \subseteq \mathcal{P}_2(V) := \{\{a,b\} \subseteq V: a\neq b\}$. We sometimes write $E(G)$ for ...

**5**

votes

**1**answer

410 views

### Bipartite Graphs arising from two k-partitions of a given Graph

Let $G$ be an $n$-chromatic connected graph. Let $(V_1, V_2, \cdots, V_n)$ and $(U_1, U_2, \cdots, U_n)$ be two partitions of $V(G)$ corresponding to proper n-colorings of $G$.
Consider the bipartite ...

**3**

votes

**1**answer

153 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.

**4**

votes

**1**answer

152 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}}: ...

**0**

votes

**1**answer

185 views

### Reverse optimization of a minimum cost flow network

Given an undirected graph $(V,E)$, with $W$ as the weight of each edge, and a convex cost function $F(X)$, such as $|X|^k$ ($k>1$).
The cost to send $x$ unit of flow through edge $e_i$ is defined ...

**1**

vote

**0**answers

39 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 $G$ ...

**7**

votes

**2**answers

193 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 ...

**4**

votes

**1**answer

86 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 ...

**4**

votes

**0**answers

121 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

218 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

72 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 ...