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Questions tagged [graph-theory]

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, topological-graph-theory, random-graphs, graph-colorings and several others.

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6answers
6k views

Which graphs are Cayley graphs?

Every group presentation determines the corresponding Cayley graph, which has a node for each group element, and arrows labeled with the generators to get from one group element to another. My main ...
62
votes
4answers
9k views

What are the implications of the new quasi-polynomial time solution for the Graph Isomorphism problem?

This week, news came out that Laszlo Babai has found a quasi-polynomial time algorithm to solve the Graph Isomorphism problem (that is: $O(\exp(polylog(n)))$). He is giving a series of talks this week,...
58
votes
9answers
6k 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 ...
53
votes
4answers
12k views

What is a chess piece mathematically?

Historically, the current "standard" set of chess pieces wasn't the only existing alternative or even the standard one. For instance, the famous Al-Suli's Diamond Problem (which remained open for more ...
52
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4answers
7k 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 ...
50
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17answers
8k views

Generalizations of the Four-Color theorem

The four color theorem asserts that every planar graph can be properly colored by four colors. The purpose of this question is to collect generalizations, variations, and strengthenings of the four ...
49
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1answer
4k views

Why are there 1024 Hamiltonian cycles on an icosahedron?

Fix one edge $e$ of the graph (1-skeleton) of an icosahedron. By a computer search, I found that there are 1024 Hamiltonian cycles that include $e$. [But see edit below re directed vs. undirected!] ...
47
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4answers
8k 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 $n\...
43
votes
2answers
4k views

Does knight behave like a king in his infinite odyssey?

The Knight's Tour is a well-known mathematical chess problem. There is an extensive amount of research concerning this question in two/higher dimensional finite boards. Here, I would like to tackle ...
43
votes
4answers
7k 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, ...
43
votes
3answers
2k views

History of the four-colour problem

It is stated in many places that the first published reference to the four-colour problem (aka the four-color problem) was an anonymous article in The Athenæum of April 14, 1860, attributed to de ...
42
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1answer
1k views

Can't one walk to infinity on the prime numbers with finitely many distinct affine steps?

Let $(a_1,b_1), \dots, (a_k,b_k)$ be finitely many pairs of positive integers, and let $\Gamma$ be the graph whose vertices are the prime numbers and in which two vertices $p$ and $q$ are connected by ...
42
votes
0answers
1k views

Does every triangle-free graph with maximum degree at most 6 have a 5-colouring?

A very specific case of Reed's Conjecture Reed's $\omega$,$\Delta$, $\chi$ conjecture proposes that every graph has $\chi \leq \lceil \tfrac 12(\Delta+1+\omega)\rceil$. Here $\chi$ is the chromatic ...
39
votes
15answers
7k views

Strengthening the Induction Hypothesis

Suppose you are trying to prove result $X$ by induction and are getting nowhere fast. One nice trick is to try to prove a stronger result $X'$ (that you don't really care about) by induction. This ...
39
votes
4answers
3k views

Why is “P vs. NP” necessarily relevant?

I want to start out by giving two examples: 1) Graham's problem is to decide whether a given edge-coloring (with two colors) of the complete graph on vertices $\lbrace-1,+1\rbrace^n$ contains a ...
39
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6answers
4k 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 ...
39
votes
1answer
2k views

What is the sandpile torsor?

Let G be a finite undirected connected graph. A divisor on G is an element of the free abelian group Div(G) on the vertices of G (or an integer-valued function on the vertices.) Summing over all ...
38
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23answers
60k views

Good programs for drawing graphs ( directed weighted graphs )

Does anyone know of a good program for drawing directed weighted graphs?
37
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17answers
10k 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, ...
37
votes
5answers
4k views

Algebraic Proof of 4-Colour Theorem?

4-Colour Theorem. Every planar graph is 4-colourable. This theorem of course has a well-known history. It was first proven by Appel and Haken in 1976, but their proof was met with skepticism ...
34
votes
21answers
4k 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; ...
34
votes
14answers
18k views

What are the Applications of Hypergraphs

Hypergraphs are like simple graphs, except that instead of having edges that only connect 2 vertices, their edges are sets of any number of vertices. This happens to mean that all graphs are just a ...
34
votes
2answers
3k views

How to find Erdős' treasure trove?

The renowned mathematician, Paul Erdős, has published more than 1500 papers in various branches of mathematics including discrete mathematics, graph theory, number theory, mathematical analysis, ...
34
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7answers
6k views

Spectral graph theory: Interpretability of eigenvalues and -vectors

I thought "Wow!" when I learned that the eigenvector of the adjacency matrix of a cycle graph $C_n$ corresponding to the second largest eigenvalue gives the coordinates of the vertices when equally ...
33
votes
4answers
1k views

Graph containing all trees?

Consider graphs on $n$ nodes. I am trying to find a graph $G$ that contains all $n$-node trees as sub-graphs but contains as few edges as possible. The complete graph $K_n$ suffices, but can we get ...
33
votes
5answers
44k views

How large is TREE(3)?

Friedman, in http://www.math.osu.edu/~friedman.8/pdf/EnormousInt112201.pdf, shows that TREE(3) is much larger than n(4), itself bounded below by $A^{A(187195)}(3)$ (where $A$ is the Ackerman ...
32
votes
18answers
14k 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 ...
32
votes
2answers
1k views

A conjecture on planar graphs

I don't know the following is a known result, but it would be very useful to me in my research if it were true. Conjecture: Let $G$ be a planar graph. The sum $$ \sum_{\{x,y\} \in E(G)}{\min(deg(x),...
31
votes
3answers
5k views

Bringing Number and Graph Theory Together: A Conjecture on Prime Numbers

Some MOers have been skeptic whether something like natural number graphs can be defined coherently such that every finite graph is isomorphic to such a graph. (See my previous questions [1], [2], [3]...
31
votes
3answers
3k views

How much linear algebra can be done with graphs?

Let G be a finite directed acyclic graph, with sources $A=\{a_1,\ldots,a_n\}$ and sinks $B=\{b_1,\ldots,b_n\}$, with edge weights $w_{ij}$. The weight of a directed path P is the product of weights of ...
31
votes
3answers
2k views

Arranging numbers from $1$ to $n$ such that the sum of every two adjacent numbers is a perfect power

I've known that one can arrange all the numbers from $1$ to $\color{red}{15}$ in a row such that the sum of every two adjacent numbers is a perfect square. $$8,1,15,10,6,3,13,12,4,5,11,14,2,7,9$$ ...
31
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1answer
1k views

What does this connection between Chebyshev, Ramanujan, Ihara and Riemann mean?

It all started with Chris' answer saying returning paths on cubic graphs without backtracking can be expressed by the following recursion relation: $$p_{r+1}(a) = ap_r(a)-2p_{r-1}(a)$$ $a$ is an ...
31
votes
0answers
972 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 = (\...
30
votes
9answers
3k 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. ...
30
votes
3answers
3k views

Can assignment solve stable marriage?

This is an excellent question asked by one of my students. I imagine the answer is "no", but it doesn't strike me as easy. Recall the set up of the stable marriage problem. We have $n$ men and $n$ ...
30
votes
5answers
2k views

Does every bipartite graph with 512 edges have an induced subgraph with 256 edges?

Suppose we have a (simple) bipartite graph with $2^k$ edges. Is it true that there is a subset of the vertices such that their induced subgraph has exactly $2^{k-1}$ edges? I know that the answer is ...
30
votes
1answer
3k views

An edge partitioning problem on cubic graphs

Hello everyone, I already asked this question on the TCS Stack Exchange, but it has not been resolved yet. Maybe readers of this forum will have other ideas or information, although I suspect that ...
29
votes
8answers
4k views

Is there a “knot theory” for graphs?

I think knot theory has been studied for quite a while (like a century or so), so I'm just wondering whether there is a "knot theory" for graphs, i.e. the study of (topological properties of) ...
29
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1answer
2k views

Should axiomatic set theory be translated into graph theory?

Recently I saw the abstract of a paper by Nash-Williams: ``Should axiomatic set theory be translated into graph theory?''. The abstract, taken from Mathscinet says the following: The author ...
28
votes
3answers
2k views

Has there been a computer search for a 5-chromatic unit distance graph?

The existence of a 4-chromatic unit distance graph (e.g., the Moser spindle) establishes a lower bound of 4 for the chromatic number of the plane (see the Nelson-Hadwiger problem). Obviously, it ...
28
votes
4answers
3k 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 ...
28
votes
3answers
1k views

Is the Rado graph a Cayley graph? If so, what is the group like? (And other questions…)

The countable random graph, also known as the Rado graph, is characterized as the unique countable graph in which every two disjoint finite sets $A$ and $B$ of vertices admit a vertex $p$ connected to ...
27
votes
9answers
3k 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 ...
27
votes
8answers
4k views

Applications of infinite graph theory

Finite graph theory abounds with applications inside mathematics itself, in computer science, and engineering. Therefore, I find it naturally to do research in graph theory and I also clearly see the ...
27
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10answers
2k views

What (fun) results in graph theory should undergraduates learn?

I have the task of creating a 3rd year undergraduate course in graph theory (in the UK). Essentially the students will have seen minimal discrete math/combinatorics before this course. Since graph ...
27
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2answers
2k 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 ...
27
votes
1answer
2k views

Does this knot invariant distinguish trefoil chiralities?

Let $C_N$ denote the labelled configuration of $N^{th}$ roots of unity with $p_J = e^{\frac{2\pi iJ}{N}}$ for $J = 1\ldots N$. As a corollary of something else I was playing around with, I recently ...
27
votes
2answers
974 views

Is this graph polynomial known? Can it be efficiently computed?

I am a physicist, so apologies in advance for any confusing notation or terminology; I'll happily clarify. To provide a minimal amount of context, the following graph polynomial came up in my research ...
26
votes
1answer
2k 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 ...
26
votes
2answers
461 views

A small unavoidable collection of subgraphs

What is the smallest number S(k,n) of unlabeled graphs on k vertices such that every simple graph on n vertices contains at least one of these as an induced subgraph? I'd like to avoid exhaustive ...