Questions tagged [graph-theory]

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.

Graph theory is the study of graphs, which is defined as an ordered pair $G = (V, E)$ comprising a set $V$ of vertices or nodes or points together with a set $E$ of edges or arcs or lines, which are 2-element subsets of V (i.e. an edge is associated with two vertices, and that association takes the form of the unordered pair comprising those two vertices).

Questions involve graph properties, graph algorithms, proofs and examples involving graphs, and applications of graph theory to other fields or practical ends.

Use instead for questions about graphing or plotting of functions.

21786 questions
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Is it possible to draw this picture without lifting the pen?

Some days ago, our math teacher said that he would give a good grade to the first one that will manage to draw this: To draw this without lifting the pen and without tracing the same line more than once. It's a bit like the "nine dots" puzzle but…
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What is that thing that keeps showing in papers on different fields?

A few months ago, when I was studying strategies for the evaluation of functional programs, I found that the optimal algorithm uses something called Interaction Combinators, a graph system based on a few nodes and rewrite rules. I've implemented…
MaiaVictor
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How do I explain the Königsberg Bridge problem to a child?

I am going to demonstrate the Königsberg seven bridge problem in a science exhibition. I am also going to use a model for a more visual representation of the problem. Now, how do I explain this (the solution) simply to a child who is not too much…
Soham
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$n$ people sitting on a circular table without repeating neighbour-sets

I made this problem up and it's been bothering me ever since. We're organising team activities in our company for the next few days. Our team consists of $n$ people seated on a circular table. To spice it up, we plan to do it in such a way that no…
P.K.
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How to calculate the number of possible connected simple graphs with $n$ labelled vertices

Suppose that we had a set of vertices labelled $1,2,\ldots,n$. There will several ways to connect vertices using edges. Assume that the graph is simple and connected. In what efficient (or if there is no efficient way, you can just tell me whatever…
Shoeinger Veronica
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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 "deeper" areas of mathematics. But then I stumbled over…
Dominik
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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 reaches the diagonally opposite vertex?
Neel
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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 the horizontal axis, and we can use this…
Jakub Konieczny
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Is Category Theory similar to Graph Theory?

The following author noted: Roughly speaking, category theory is graph theory with additional structure to represent composition. My question is: Is Category Theory similar to Graph Theory?
hawkeye
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Blocking directed paths on a DAG with a linear number of vertex defects.

Let $G=(V,E)$ be a directed acyclic graph. Define the set of all directed paths in $G$ by $\Gamma$. Given a subset $W\subseteq V$, let $\Gamma_W\subseteq \Gamma$ be the set of all paths $\gamma\in\Gamma$ supported on $V\backslash W$ (i.e all…
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Connecting three houses to three utilities

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 over each other (wires can't meet and there is no…
M.S.E
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Second-largest connected component of mathematicians

Let $\Gamma$ be the graph defined as follows: the vertices are mathematicians who have published papers there is an edge between any two mathematicians who have authored publications together. Graph $\Gamma$ is certainly not connected, as there…
Gaussler
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A disease spreading through a triangular population

I have run into this problem in my research, which I'm presenting under a different guise to avoid going into unnecessary background. Consider a population that is connected in a triangular manner, as shown in the figure below. The root node has a…
MGA
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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 $p$. Let $z(t)$ denote the number of infected nodes…
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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, graph two has $2$ simple circuits of length $3$…
dukevin
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