**-1**

votes

**1**answer

221 views

### an interesting conjecture about even cycle

Let G be a simple graph which is a $2n$-cycle equipped with $n$ chords such that $G$ is $3$-regular,in other words,the set of the $n$ chords is a perfect matching of $G$(that is,every vertex of $G$ is ...

**14**

votes

**8**answers

2k views

### Representability of finite metric spaces

There have been a couple questions recently regarding metric spaces, which got me thinking a bit about representation theorems for finite metric spaces.
Suppose $X$ is a set equipped with a metric ...

**5**

votes

**1**answer

462 views

### Difference Sets

Suppose
$$
P \subseteq \{1,2,\dots,N\},\quad |P| = K
$$
We calculate the differences as: $$d=p_i-p_j\mod N,\quad i\ne j$$
Now let $a_d$ denote the number of occurrence of $d$ (for $d = 1, 2, \dots , N ...

**43**

votes

**8**answers

4k views

### What is a continuous path?

I would like some help, because I am getting mad trying to answer the following
Question: Let $X$ be a topological space, what is a continuous path in $X$?
Well, maybe you're already getting ...

**29**

votes

**20**answers

3k views

### Generalizations of Planar Graphs

This is a follow up to Harrison's question: why planar graphs are so exceptional. I would like to ask about (and collect answers to) various notions, in graph theory and beyond graph theory (topology; ...

**25**

votes

**9**answers

2k views

### How many groups of size at most n are there? What is the asymptotic growth rate? And what of rings, fields, graphs, partial orders, etc.?

Question. How many (isomorphism types of) finite groups of size at most n are there? What is the asymptotic growth rate? And the same question for rings,
fields, graphs, partial orders, etc.
...

**24**

votes

**4**answers

2k views

### Adjacency matrices of graphs

Motivated by the apparent lack of possible classification of integer matrices up to conjugation (see here) and by a question about possible complete graph invariants (see here), let me ask the ...

**6**

votes

**1**answer

987 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 ...

**15**

votes

**2**answers

778 views

### Minimal graphs with a prescribed number of spanning trees

As its long ago since Erdős died and mathoverflow is the second best alternative to him (for discussing personal problems), I'd like to start a fruitful discussion about the following problem ...

**9**

votes

**2**answers

671 views

### Is the Steiner ratio Gilbert–Pollak conjecture still open?

Gilbert-Pollak conjecture on the Steiner ratio: Consider a set $P$ of $n$ points on the euclidean plane. A shortest
network interconnecting $P$ must be a tree, which is called a Steiner minimum ...

**4**

votes

**4**answers

984 views

### Delaunay triangulations and convex hulls

This is a reference request.
I have the impression that those who work in computational geometry are accustomed to the following. You have some locally finite set of sites in $\mathbb{R}^n$ and you ...

**4**

votes

**1**answer

100 views

### genus of a finite simple undirected graph

Let $G$ be a finite simple undirected graph. Suppose there exist subgraphs $G_1,G_2,\dots,G_n$ of $G$, such that $G_i$ and $G_j$, have no common edges and have at most two common vertices, for each ...

**3**

votes

**1**answer

1k views

### Counting graphs

I'd love your help with this question.
Let $n\geq3$ be a fixed integer. How many non-isomorphic graphs with $p$ vertices and $q$ edges are there where $p+q=n$?
Thank you very much.

**5**

votes

**2**answers

313 views

### Generalisations of Petersen's 2-factor theorem?

Petersen's 2-Factor Theorem (1891): A $(2r)$-regular graph can be decomposed into $r$ edge-disjoint $2$-factors.
I'd like to use this theorem (or a more general version of this theorem) to imply ...

**4**

votes

**1**answer

181 views

### Graphs with constant edge imbalance

The imbalance of an edge $(u,v) \in E(G)$ of a graph $G$ is defined as $|d(u)-d(v)|$ ($d$ being, as usual the degree). (This concept was introduced by Albertson in 1997)
I'm interested in the set of ...

**2**

votes

**1**answer

193 views

### Min Bend Orthogonal Knots

I am seeking literature on 3D orthogonal drawings of knots,
especially minimum bend drawings.
An orthogonal drawing employs segments parallel to the axes of
a Cartesian coordinate system.
A bend is a ...

**1**

vote

**0**answers

128 views

### Non-trivial lower bound on the number of “Graph Diagonals”

The definition of Graph Diagonals, that are the subject of this question, is based on the notions of crossing edges and on connected graphs:
Two edges $AC$ and $BD$ of a complete, symmetric and ...

**23**

votes

**15**answers

6k 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, ...

**28**

votes

**3**answers

4k views

### Why are planar graphs so exceptional?

As compared to classes of graphs embeddable in other surfaces.
Some ways in which they're exceptional:
Mac Lane's and Whitney's criteria are algebraic characterizations of planar graphs. (Well, ...

**39**

votes

**4**answers

6k views

### Do there exist chess positions that require exponentially many moves to reach?

By "chess" here I mean chess played on an $n\times n$ board with an unbounded number of (non-king) pieces. Some care is needed if you want to generalize some of the subtler rules of chess to an ...

**23**

votes

**18**answers

6k views

### Interesting and Accessible Topics in Graph Theory

This summer, I will be teaching an introductory course in graph theory to talented high school seniors. The intent of the course is not to establish proficiency in graph theory, per se. Rather, I hope ...

**17**

votes

**5**answers

2k views

### Complete graph invariants?

Obviously, graph invariants are wonderful things, but the usual ones (the Tutte polynomial, the spectrum, whatever) can't always distinguish between nonisomorphic graphs. Actually, I think that even a ...

**24**

votes

**3**answers

620 views

### Removal of non-isomorphic edges results in the same graph

There exists a (simple unlabeled) graph on 6 nodes with a pair of non-isomorphic edges (i.e., there is no graph automorphism that sends one edge into the other) such that removal of either of them ...

**10**

votes

**1**answer

681 views

### Different uses of the word “ergodic”

There appear to be two definitions of the word ergodic.
The dynamical systems definition says that a measure space $(X,\mathit B, \mu)$ and measure preserving transformation $T: X \mapsto X$ is ...

**16**

votes

**1**answer

674 views

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

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 numbers ...

**40**

votes

**4**answers

3k views

### Connectivity of the Erdős–Rényi random graph

It is well-known that if $\omega=\omega(n)$ is any function such that $\omega \to \infty$ as $n \to \infty$, and if $p \ge (\log{n}+\omega) / n$ then the Erdős–Rényi random graph $G(n,p)$ is ...

**11**

votes

**10**answers

1k views

### Chromatic number of graphs of tangent closed balls

The Koebe–Andreev–Thurston theorem gives a characterization of planar graphs in terms of disjoint circles being tangent. For every planar graph $G$ there is a disk packing whose graph is $G$. What ...

**9**

votes

**3**answers

1k views

### The diameter of the Erdös component of the collaboration graph

This site claims that the diameter of the Erdös component of the collaboration graph in 2004 was 23. What is it now? Is it increasing or decreasing with time? Recall that the vertices of the ...

**34**

votes

**6**answers

3k views

### Is it easy to produce hard-to-color graphs?

This question arises from my recent visit to my daughter's second-grade class, where I led some discussion and activities on graph coloring (see Math for seven-year-olds). In one such activity, each ...

**23**

votes

**9**answers

2k views

### Is the empty graph a tree?

This is a boring, technical question that I stumbled upon while making a contribution to Sage. I would still like to hear a constructive answer so hopefully the question does not get closed.
The ...

**20**

votes

**1**answer

1k views

### Monochromatic triangles in every two-coloring of the plane?

An old problem (possibly due to Erdős and Graham?): given a triangle $T$ and a two-coloring of the plane, does there necessary exist a monochromatic congruent copy of $T$? Here "monochromatic" means ...

**13**

votes

**1**answer

724 views

### Is there a group whose cardinality counts non-intersecting paths?

Introduction
Graphs are not only important combinatorial objects, but also related to many topological/algebraic structures. In this question I am going to talk about various group structures with ...

**10**

votes

**3**answers

1k views

### Is non-connectedness of graphs first order axiomatizable?

A recent
question
asked for graph properties that are first order axiomatizable but not finitely axiomatizable.
Connectedness was mentioned in the context. Connectedness can be axiomatized in ...

**14**

votes

**1**answer

644 views

### Is every graph the center of some other graph?

The center of a graph $G$ is the set of vertices that minimize the largest
distance to vertices in $G$, e.g., in the graph below, that radius is $4$:
Define the ...

**14**

votes

**3**answers

527 views

### Complexity of equitable partitions

We are talking about undirected simple graphs and partitions of their vertex sets into disjoint non-empty cells. Such a partition is equitable if for any two vertices $v,w$ in the same cell, and any ...

**13**

votes

**2**answers

1k views

### Realizing groups as automorphism groups of graphs.

Frucht showed that every finite group is the automorphism group of a finite graph. The paper is here.
The argument basically is that a group is the automorphism group of its (colored) Cayley graph
...

**9**

votes

**6**answers

959 views

### Where on the internet I can find database of graphs?

I am studying graph algorithms.
I need database of graphs on which I can test my algorithms.
Where can I find reliable database of graphs of all kinds?
Thanks!

**14**

votes

**0**answers

584 views

### A Conjecture About Directed Graphs that are the Union of Two Trees

Let D=(V,E) be a directed graph that is the union of two edge-disjoint directed
spanning trees. Suppose that
There no subset X of vertices so that
there is precisely one directed edge
from X ...

**13**

votes

**6**answers

4k views

### Good algorithm for finding the diameter of a (sparse) graph?

My question on Stack Overflow was recently tagged "math". Despite a bounty, it never received a satisfactory answer, so I thought I would ask it here:
I have a large, connected, sparse graph in ...

**12**

votes

**4**answers

911 views

### Graphs in which every spanning tree is an independency tree

It follows from this question
and the corresponding answers, that the complete graphs and the cycles are precisely the graphs
$G$ having the property that, for every spanning tree $T$ of $G$, the ...

**7**

votes

**2**answers

438 views

### Is there an algorithm to compute efficiently the dessin d'enfant from a Belyi pair?

Let $(X,f)$ be a Belyi pair, i.e. a Riemann surface $X$ together with a morphism $f: X \to \mathbb{P}^1$, ramified only in $0,1, \infty$. Grothendieck's dessin d'enfant is the pre-image $G$ of the ...

**7**

votes

**3**answers

1k views

### On “super connected” graphs

A graph $G$ is called super connected if for every connected subgraph $H\subset G$ the graph $G-H$ obtained from $G$ after deletion of all vertices from $H$ is also connected.
Conjecture: The only ...

**5**

votes

**1**answer

1k views

### Non-negative quadratic maximization

For a given symmetric and positive semidefinite $n \times n$ matrix $A$, we want to solve the problem $$\max_{||x|| = 1, \ x\geq 0} x^T A x.$$
Here, $x\geq 0$ indicates that $x$ must be component-wise ...

**3**

votes

**2**answers

441 views

### spectrum of an adjacency matrix

The adjacency matrix of a non-oriented connected graph is symmetric, hence its spectrum is real.
If the graph is bipartite, then the spectrum of its adjacency matrix is symmetric about 0. A few ...

**3**

votes

**1**answer

198 views

### A question on graphic sequences

Let $G$ be a graph and $d_{G}(u)$ denotes degree of a vertex $u$ in $G$. Consider the next multiset $$M_{G}:=\{|d_{G}(u)-d_{G}(v)|:\ uv\in E(G)\}.$$
Conjecture: $M_{G}$ is graphical for every $G$.
...

**15**

votes

**1**answer

614 views

### Paul Erdős: Determine or estimate the number of maximal triangle-free graphs on n vertices

Among the collections of the open problems of Paul Erdős on the website of
Professor Fan Chung, there is one called "number of triangle-free graphs".
...

**13**

votes

**1**answer

624 views

### Who first dubbed them “expander graphs”?

Expander graphs
("sparse graphs that have strong connectivity properties")
burst onto the mathematical scene around the millennium, but I have not
been successful in tracing the origin of
(a) the ...

**12**

votes

**2**answers

1k views

### Determinants in Graph Theory

In graph theory, we work with adjacency matrices which define the connections between the vertices. These matrices have various properties in themselves. For example, their trace can be calculated (it ...

**9**

votes

**3**answers

885 views

### Classification of degree (bi-)sequences of bipartite graphs?

It is known that the sequence $d_1 \geq d_2 \geq \ldots \geq d_n$ of nonnegative integers is the degree sequence of a graph if and only if the sum of the $d_i$ is even and we have
\[
\sum_{i = 1}^k ...

**8**

votes

**4**answers

2k views

### How many $p$-regular graphs with $n$ vertices are there?

Suppose that there are $n$ vertices, we want to construct a regular graph with degree $p$, which, of course, is less than $n$. My question is how many possible such graphs can we get?