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
votes
1answer
212 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 ...
13
votes
8answers
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
1answer
453 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
8answers
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 ...
23
votes
9answers
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. ...
23
votes
4answers
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
1answer
959 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
2answers
761 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
2answers
592 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
4answers
910 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
1answer
98 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 ...
4
votes
1answer
179 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
1answer
188 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
0answers
125 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
15answers
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, ...
27
votes
3answers
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, ...
23
votes
18answers
5k 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 ...
37
votes
3answers
5k 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 ...
29
votes
20answers
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; ...
15
votes
5answers
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 ...
10
votes
1answer
651 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
1answer
630 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 ...
37
votes
4answers
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 ...
9
votes
3answers
949 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 ...
33
votes
6answers
2k 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
9answers
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
1answer
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
1answer
712 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
3answers
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
3answers
505 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 ...
14
votes
0answers
569 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
1answer
634 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 ...
12
votes
4answers
905 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 ...
12
votes
6answers
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 ...
7
votes
3answers
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 ...
6
votes
2answers
416 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 ...
5
votes
1answer
926 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
2answers
334 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
1answer
192 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
1answer
564 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
1answer
594 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
2answers
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
3answers
799 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
4answers
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?
5
votes
1answer
581 views

Algorithm for embedding a graph with metric constraints

Suppose I have a graph $G$ with vertex set $V$, edge set $E \subseteq {V \choose 2}$, and a weight function $w:E \to \mathbb{R}^{+}$. Is there a nice algorithmic way to decide if there is an ...
4
votes
0answers
211 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 ...
3
votes
2answers
235 views

Can the Vertices of cubic graph be partitioned into and induced cycle and a forest?

Let $G$ be a $2$-connected $3$-regular graph. Can $V(G)$ be partitioned into $V_1$ and $V_2$ where $G[V_1]$(the induced subgraph on $V_1$) is a cycle of $G$ and $G[V_2]$ is a forest (Acyclic ...
3
votes
1answer
358 views

Path cardinality for random $(a+b)$-ary infinite trees

Consider a random infinite binary tree $T(a,b)$, so that $a$ denotes the probability of a left edge branching from any root-connected node,and $b$ denotes the probability of a right edge branching ...
11
votes
5answers
2k views

Are there more connected or disconnected graphs on n vertices?

Suppose we are talking about graphs with $n$ labeled vertices. Which graphs are more common: connected or non connected?
9
votes
1answer
345 views

Uniquely hamiltonian graphs with minimum degree 4

A graph is uniquely hamiltonian if it has exactly one Hamilton cycle. As every edge in a cubic graph lies in an even number of Hamilton cycles, a cubic graph cannot be uniquely hamiltonian, and a ...