Use this tag for questions in graph theory. Here a graph is a collection of vertices and connecting edges. Use (graphing-functions) instead if your question is about graphing or plotting functions.

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77
votes
1answer
2k views

Number of simple edge-disjoint paths needed to cover a planar graph

Let $G=(V,E)$ be a graph with $|E|=m$ of a graph class $\mathcal{G}$. A path-cover $\mathcal{P}=\{P_1,\ldots,P_k\}$ is a partition of $E$ into edge-disjoint simple paths. The size of the cover is ...
49
votes
15answers
53k views

Online tool for making graphs (vertices and edges)?

Anyone know of an online tool available for making graphs (as in graph theory - consisting of edges and vertices)? I have about 36 vertices and even more edges that I wish to draw. (why do I have so ...
39
votes
2answers
742 views

Counting trails in a triangular grid

A triangular grid has $N$ vertices, labeled from 1 to $N$. Two vertices $i$ and $j$ are adjacent if and only if $|i-j|=1$ or $|i-j|=2$. See the figure below for the case $N = 7$. How many trails ...
37
votes
4answers
2k views

How does the divisibility graphs work?

I came across this graphic method for checking divisibility by $7$. $\hskip1.5in$ Write down a number $n$. Start at the small white node at the bottom of the graph. For each digit $d$ in ...
35
votes
6answers
2k views

Motivation for spectral graph theory.

Why do we care about eigenvalues of graphs? Of course, any novel question in mathematics is interesting, but there is an entire discipline of mathematics devoted to studying these eigenvalues, so ...
34
votes
1answer
599 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. $$8,1,15,10,6,3,13,12,4,5,11,14,2,7,9$$ ...
33
votes
0answers
1k views

Connecting three houses to three utilities [duplicate]

When I was a child I was given this problem to send a wire from electricity, water, and internet to each of the houses, all three houses must have all three wires connected without being crossed ...
31
votes
16answers
17k views

What are good books to learn graph theory?

What are some of the best books on graph theory, particularly directed towards an upper division undergraduate student who has taken most the standard undergraduate courses? I'm learning graph theory ...
29
votes
1answer
2k views

Is Wolfram wrong about unique 3-colorability, or am I just confused?

The illustration on Wolfram's page claims to present a uniquely colorable, triangle-free graph. However, this seems to be blatantly false: the graph has a symmetry with respect to a reflection through ...
28
votes
1answer
550 views

Zombie outbreak on a $k$-regular graph

Suppose we have a zombie outbreak on a connected $k$-regular graph of order $n$. There are $n_0$ initially infected zombie nodes, and each turn, each zombie infects its neighbors with probability ...
27
votes
1answer
5k views

What do the eigenvectors of an adjacency matrix tell us?

The principal eigenvector of the adjacency matrix of a graph gives us some notion of vertex centrality. What do the second, third, etc. eigenvectors tell us? Motivation: A standard information ...
26
votes
1answer
10k views

Is Sage on the same level as Mathematica or Matlab for graph theory and graph visualization?

The context: I'm going to start working on a project that involves running predefined algorithms (and defining my own) for very big graphs (thousands of nodes). Visualization would also be welcome if ...
25
votes
3answers
1k views

An example of a real-world map that is not 4-colourable?

The four-colour mapping theorem states that all maps can be four-coloured (adjacent regions receive distinct colours, and four different colours are used in total). However, the technical definition ...
25
votes
4answers
632 views

Is this graph connected

Define the following graph on the vertex set ${\mathbb N}_{\geq1}\>$: Two numbers $a$, $b\in {\mathbb N}_{\geq1}$ are connected by an edge (written $a \ \mathcal{R} \ b)$ if and only if $a+b \ | ...
24
votes
1answer
167 views

Smallest graph with automorphism group the quaternion $8$-group, $Q_8$

Frucht's Theorem states that for any finite group $G$ there is a finite (undirected) graph $\Gamma$ for which the automorphism group $\text{Aut}(\Gamma)$ of $\Gamma$ is isomorphic to $G$, and for many ...
24
votes
0answers
523 views

Is Erdős' lemma on intersection graphs a special case of Yoneda's lemma?

Under which name is the following proposition filed actually: Every poset $P$ embeds fully and faithfully in the powerset of $P$, ordered by subset inclusion. Let me call it Dedekind's lemma. ...
22
votes
4answers
5k views

Are these 2 graphs isomorphic?

They meet the requirements of both having an $=$ number of vertices ($7$). They both have the same number of edges ($9$). They both have $3$ vertices of degree $2$ and $4$ of degree $3$. However, ...
22
votes
2answers
890 views

Does Birkhoff - von Neumann imply any of the fundamental theorems in combinatorics?

I recently had the occasion to think about Hall's Marriage Theorem for the first time since my undergraduate combinatorics class more than a decade ago. Reading the wikipedia article linked above, I ...
21
votes
1answer
667 views

How to create mazes on the hyperbolic plane?

I'm interested in building maze-like structures on the [5, 4] tiling of the hyperbolic plane, where by maze-like I mean something akin to a spanning tree of the underlying lattice: a subgraph of the ...
20
votes
4answers
7k views

Logic question: Ant walking a cube

There is a cube and an ant is performing a random walk on the edges where it can select any of the 3 adjoining vertices with equal probability. What is the expected number of steps it needs till it ...
20
votes
3answers
1k views

Why should “graph theory be part of the education of every student of mathematics”?

Until recently, I thought that graph theory is a topic which is well-suited for math olympiads, but which is a very small field of current mathematical research with not so many connections to ...
20
votes
3answers
1k views

Exceptional books on real world applications of graph theory.

What are some exceptional graph theory books geared explicitly towards real-world applications? I would be interested in both general books on the subject (essentially surveys of applied graph ...
20
votes
1answer
1k views

The $n$ Immortals problem.

I saw this riddle posted on reddit a long time ago, called the "Seven Immortals." In the beginning, the world is inhabited by seven immortals, ageless and sexless, who begin to multiply and ...
20
votes
3answers
632 views

Math puzzle: 10 digit strings generations

There was a question in a math competition that I attended last year. At the end of competition, I realized that my answer was wrong for the question below and I have never been able to figure out how ...
19
votes
11answers
9k views

Graph theory software?

Is there any software that for drawing graphs (edges and nodes) that gives detailed maths data such as degree of each node, density of the graph and that can help with shortest path problem and with ...
19
votes
2answers
829 views

Cover time chess board (king)

Consider a random walk of a king on a standard chess board, which at each step moves to a uniformly random permitted square. What's the exact mean time to visit all squares (cover time), starting ...
19
votes
2answers
598 views

Connecting a $n, n$ point grid

I stumbled across the problem of connecting the points on a $n, n$ grid with a minimal amount of straight lines without lifting the pen. For $n=1, n=2$ it is trivial. For $n=3$ you can find the ...
18
votes
2answers
885 views

Not lifting your pen on the $n\times n$ grid

Does there exist $n$, and $r<2n-2$, such that the $n\times n$ square grid can be connected with an unbroken path of $r$ straight lines? Note: This has essentially already been asked - see this ...
18
votes
2answers
358 views

Generalized nontransitive dice

Let $X_1, \ldots, X_n$ be a collection of random variables. Consider the directed graph with vertex set $\{ 1, 2, \ldots, n \}$ where there is a directed edge $i \to j$ if $\mathbb{P}(X_i > X_j) ...
18
votes
2answers
499 views

Groups and generating sets

This question feels completely trivial and I am somewhat embarrassed to be asking it, but I am having a brain dead moment and failing to prove what I'm sure is a completely trivial statement about ...
18
votes
2answers
2k views

Human checkable proof of the Four Color Theorem?

Four Color Theorem is equivalent to the statement: "Every cubic planar bridgeless graphs is 3-edge colorable". There is computer assisted proof given by Appel and Haken. Dick Lipton in of his ...
18
votes
2answers
883 views

Self-avoiding walk on $\mathbb{Z}$

How many sequences $a_1,a_2,a_3,\dotsc$, satisfy: i) $a_1=0$ ii) ($a_{n+1}=a_n-n$ or $a_{n+1}=a_n+n$) iii) $a_i\neq a_j$ for $i\neq j$ iiii) $\mathbb{Z}=\{a_i\}_{i>0}$ Are the two alternating ...
17
votes
2answers
297 views

A game on a graph

Alice and Bob play a game on a complete graph ${G}$ with $2014$ vertices. They take moves in turn with Alice beginning. At each move Alice directs one undirected edge of $G$. At each move Bob chooses ...
17
votes
1answer
230 views

Bombing of Königsberg problem

A well-known problem in graph theory is the Seven Bridges of Königsberg. In Leonhard Euler's day, Königsberg had seven bridges which connected two islands in the Pregel River with the mainland, laid ...
17
votes
3answers
339 views

Results in graph theory proved using other areas of math, and vice versa

I'm curious about learning graph theory, as it seems to pop up in some unexpected places. In order to get a partial feel for the subject, I was wondering if anyone could point me to some survey ...
16
votes
3answers
1k views

Why do graph degree sequences always have at least one number repeated? [duplicate]

Why do graph degree sequences always have at least one number repeated? $(1, 2, 2, 3)$ = Valid, as you can see, because the $2$ is repeated. $(1, 2, 3)$ = Not possible to construct a graph with ...
16
votes
1answer
1k views

Did the Appel/Haken graph colouring (four colour map) proof really not contribute to understanding?

I hope this isn't off topic - sorry if I'm wrong. In 1976, Kenneth Appel and Wolfgang Haken proved the claim (conjecture) that a map can always be coloured with four colours, with no adjacent regions ...
16
votes
5answers
3k views

What's the relation between topology and graph theory

I read the Wikipedia articles for both topology, graph theory (plus topological graph theory). Does topology encompass also graph theory? Or topology is only about studying shapes while graph theory ...
16
votes
3answers
1k views

Why there are $11$ non-isomorphic graphs of order $4$?

I'm new to graph theory and I don't plan to become a serious student of graph theory either. My book suggests that there are $11$ non-isomorphic graphs of order $4$, but I can't see why. I know that ...
16
votes
3answers
521 views

Counting the number of polygons in a figure

I often come across figures like this on the net, or as contest problems, asking to find the number of a specific type of polygon in the figure (triangles, in this case). But I've never really found ...
16
votes
2answers
563 views

If $G$ is biconnected and $\delta(G) \geq 3 \Rightarrow \exists v: G-v$ is also biconnected.

If $G$ is biconnected and $\delta(G) \geq 3 \Rightarrow \exists v: G-v$ is also biconnected. Where $\delta (G) - $ minimum degree of all vertices, $G-v$ is equal to if we remove this vertex from $G$ ...
15
votes
3answers
4k views

Average Scrabble graph structure: diameter?

Tonight a game of Scrabble ended in what I consider a very unusual graph structure, unlike this generic web image, which seems more typical: ...
15
votes
3answers
6k views

Difference between graph homomorphism and graph isomorphism

I am still not getting how graph isomorphism and homomorphism differ. Can anyone show me two graphs that are homomorphic, but not isomorphic? Also, Wikipedia says that when the function and the ...
15
votes
3answers
417 views

Is it possible to uniquely number faces of a hexagonal grid with consecutive numbers?

You have a grid of regular hexagons. The aim of the game is to have each hex contain the numbers 1-6 on its edges. Each edge must also be connected to another edge that has a value one higher and ...
15
votes
1answer
397 views

What are all conditions on a finite sequence $x_1,x_2,…,x_m$ such that it is the sequence of orders of elements of a group?

My Definition: The finite sequence $x_1,x_2,...,x_m$ of nonnegative integers is said to be generated by a finite group $G$ iff $n:=|G|=x_1+x_2+\cdots+x_m$. $n$ has $m$ divisors. if ...
15
votes
2answers
541 views

What is the probability that every pair of students studies together at some point?

A cohort in a school consists of 75 students who study for 6 years. Each year, the students are randomly distributed into 3 classrooms of 25 students each. What is the probability that, after 6 ...
15
votes
3answers
1k views

In how many ways we can place $N$ mutually non-attacking knights on an $M \times M$ chessboard?

Given $N,M$ with $1 \le M \le 6$ and $1\le N \le 36$. In how many ways we can place $N$ knights (mutually non-attacking) on an $M \times M$ chessboard? For example: $M = 2, N = 2$, ans $= 6$ $M = 3, ...
15
votes
0answers
474 views

Normalizers of automorphism groups

In abstract groups $\Gamma$ the normalizer $N_\Gamma(S)$ of a subset $S\subseteq\Gamma$ is the subgroup of all $x \in \Gamma$ that commute with $S$, i.e. $xS = Sx$, i.e. $x\ y\ x^{-1} \in S $ for all ...
14
votes
6answers
971 views

Can an algorithm be part of a proof?

I am an undergraduate student. I have read several papers in graph theory and found something may be strange: an algorithm is part of a proof. In the paper, except the last two sentences, other ...
14
votes
3answers
362 views

Integer sequences which quickly become unimaginably large, then shrink down to “normal” size again?

There are a number of integer sequences which are known to have a few "ordinary" size values, and then to suddenly grow at unbelievably fast rates. The TREE sequence is one of these sequences, which ...