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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1answer
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Intuition behind this algorithm for finding an Eulerian circuit in a graph?

An Eulerian circuit of a directed graph $G = (V,E)$ is a path that travels through every edge in $E$ exactly once. This algorithm finds such a circuit if it exists. (I am interested in the directed ...
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3answers
53 views

Are there any examples of Graphs in nature?

When it comes to fractals, there are several examples we can point to and say 'this is a fractal', such as snowflakes, ferns, trees and coastlines. Are there any equally clear examples of graph and ...
2
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1answer
35 views

A Puzzle on Infinity: How to guess the color of hats? [duplicate]

Infinitely many (i.e. $\omega$ - many) people each have either a white hat or black hat on their heads. Each person can see everyone's hats except their own. Each person simultaneously announces a ...
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2answers
19 views

Constructing Turan Graphs

A "Turan Graph " on $n$ vertices is graph on $n$ vertices without triangles and with exactely $\lfloor \frac{n^2}{4}\rfloor$ edges. How many are the Turan Graphs on $8$ vertices? There's an easy ...
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0answers
12 views

Matching vertices between two graphs

I have a situation where I have two graphs that are supposed to represent the same underlying topology but represent the underlying topology at different resolutions. My goal is to match vertices ...
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2answers
30 views

Prove or dis-prove that it always holds or not $\lambda(G) \leq \chi(G) $ [on hold]

I want to prove that this inequality holds or not? The inequality is $\lambda(G) \leq \chi(G) $ where $\lambda(G)$ is the minimum number of edges whose deletion from a graph $G$ disconnects $G$, ...
1
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1answer
25 views

How does the following graph have an Euler tour and not every node has degree that is even?

The theorem states: A connected graph has an Euler tour if and only if every vertex has even degree. But this graph has node 'A' with degree = 3. Graph image. ...
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1answer
22 views

Show that $C_n\times K_2$ is $1$-factorable for $n\ge4$

Show that $C_n\times K_2$ is $1$-factorable (has a perfect matching) for $n\ge4.$ $\times$ means the Cartesian product. $C_n$ means a cycle where $n=$ number of vertices of the cycle. $K_2$ means the ...
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3answers
31 views

Any undirected graph on 9 vertices with minimum degree at least 5 contains a subgraph $K_4$?

Let $G$ be simple undirected graph with degree of every vertices is at least 5. Prove or disprove that $G$ contains subgraph $K_4$. I came up with this question when I were trying to find Ramsey ...
0
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3answers
18 views

Number of arcs in undirected graph

It is a basic question in graph theory! I have n nodes and I would like to calculate the number of paths among n nodes so that each node appears once in a path. I think it is Hamilton cycle, but I am ...
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0answers
20 views

Number of nodes satisfying a certain property on a binary tree

Fix a large integer $M$ and construct a binary tree as follows. Assign the root node by the integer $0$. If a node is assigned the integer $n$ and $n \leq M - 2$, then $n$ has two children and ...
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1answer
28 views

How to find List Chromatic Number of planar graphs [on hold]

I want to know how we can find the list chromatic number of planar graphs, Suppose we have graph $G= K_{3}$. Then its chromatic number is $3$, but what is the list chromatic number of $K_{3}$? ...
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0answers
23 views

Chinese Postman Problem - open walk variation

Consider the following variation of the Chinese postman problem (also known as the route inspection problem). Instead of finding the shortest closed walk which traverses each edge at least once, find ...
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1answer
32 views

Neighbourhood set in Graph theory [on hold]

Let $G$ be any connected graph with $\Delta(G)$ be maximum degree. If $D \subseteq V(G)$ then how can we say that $\left | \bigcup \limits_{v \in D} N(v) \right | \leq |D| \Delta(G)$.
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1answer
33 views

Does the complete graph contain the maximum number of simple cycles?

Let $\mathcal{G}(n,m)$ be the set of connected, simple graphs with $n$ vertices and $m$ edges. For any graph $G$ we denote its number of simple cycles with $\mu(G)$ and and for any finite family of ...
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0answers
13 views

Relationship between ST-Path Ideals and ST-Cut Ideals?

Topic: st-connectivity, st-cut ideals and path ideals of a graph My Lemma: None of the st-cut-monomials vanish iff there is at least a st-path that does not vanish. Example ST-cuts: {{1,3,5,6},{...
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0answers
46 views

relationship between uniform Hypergraph maximum matching minimum vertex cover minimum clique partition

A k-$\bf{uniform}$ hypergraph $H = (V,E)$ consists of a set $V = \{v_1, v_2, \cdots, v_n\}$ of vertices and a set $E = \{e_1, e_2, \cdots , e_m\}$ of edges, each being a size $k$ subset of $V$. (Note: ...
0
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1answer
15 views

Class of directed graph for which there is only one path to a given parent?

From a nomenclature standpoint, I am wondering if there is a name for a class of directed graph that has only one path to any given parent. I can visualize this shape as an upside down tree that may ...
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0answers
27 views

Proof of Petersons theorem (less than 3 bridges) with Tutte's Theorem

Petersons Theorem: A 3-regular graph with at most 2 bridges has a perfect matching. My task is to prove this theorem by just using Tutte and not Tutte-Berge. My first general question: Are you ...
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0answers
8 views

Acyclic orientation of a mixed graph with minimization of the critical path

I already asked this question as a guest but I was not able to edit it or add comments after I registered with my e-mail address. A apologize for asking the same question again. A mixed graph is a ...
1
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1answer
24 views

Find number of vertices when given number of edges

A Simple Connected Graph G has $M$ vertices and 4 edges, find $M$ Now lets say we didn't have any more info than what's mentioned above. By drawing out a couple of graphs I know that $M$ could ...
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1answer
22 views

Is the maximum size of a matching of graph equal to the maximum size of a matching of its dual?

This is really puzzling me! A hypergraph $H = (V,E)$ consists of a set $V = \{v_1, v_2, \cdots, v_n\}$ of vertices and a set $E = \{e_1, e_2, \cdots , e_m\}$ of edges, each being a subset of $V$. A ...
1
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1answer
29 views

If deg$(v) \geq k$ for all $v \in V(G)$, then G contains a matching of cardinality $\min \{k,\lfloor{\frac{|V|}{2}}\rfloor\}$

Let$G = (V; E)$ be an undirected graph. Show that if deg$(v) \geq k$ for all $v \in V$, then G contains a matching of cardinality $\min \{k,\lfloor{\frac{|V|}{2}}\rfloor\}$. I have no idea how to ...
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2answers
46 views

Prove that a graph has a cycle of length no more than $14$

A graph contains $2016$ vertices, its chromatic number is $5$, prove that this graph has a cycle of length $\leq 14$. Where do I start?
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0answers
16 views

Duality between cut ideals and cycle ideals?

There exist a general duality between vertex-cuts and cycles and also Duality Principle on Digraphs. I am trying to find a duality prienciple expressed in terms of ideals so Does there exist a ...
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0answers
16 views

Principle of Duality on digraphs: dual properties?

Given an arc $uv$ of a digraph $D$, the dual $D'$ of the digraph $D$ has the arc $vu$. I am trying to find dual properties for digraphs. I could find a page 301 of document on Principle of Duality for ...
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0answers
37 views

How do I write a function that maps a variable to a set?

I have a function $\Gamma$ that maps elements from $N$ to a (possibly empty) subset of $N$. The number of elements in the resulting subset depends on which element of $N$ we are dealing with, i.e. $\...
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0answers
15 views

Citations for the proof of universality of graph classes

In Automorphisms of graphs, Peter J. Cameron mentioned following classes of graphs which are universal structures. graphs of valency k for any fixed k > 2; bipartite graphs; strongly regular graphs; ...
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0answers
12 views

Orient edges in a mixed graph to minimize the critical path

3 down vote favorite A mixed graph is a graph that has directed and undirected edges. Is there an efficient algorithm that allows the orientation of undirected edges in a mixed graph in such a way ...
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0answers
36 views

How can i prove that the wiener index of a truncated wheel graph is 3n(5n - 6) where n is greater than or equal to 4? [on hold]

A truncated wheel graph TW_n, n is greater than or equal to 3, is the graph with vertex set V(TW_n) = {v_0, v_{i,1}, v_{i,2}, v_{i,3} | i = 1, 2,..., n} and edge set E(TW_n) = {v_0 v_{i,1}, v_{i,1}v_{...
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0answers
15 views

Different formulation of this problem? Packing subsets into $k$ parts.

I am currently working on the following problem which I would like to formulate in a different way to see if any work on this has been done. Let $S = \{1, 2, \dots, n\}$ be a set and $H = \{h_1, h_2, ...
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0answers
16 views

Demonstration of Cycle-cut duality on elementary graphs?

I want to see examples on the Duality theorem between cycles and cuts on the page 26 of Graph Theory Electronic Edition 2005 by Reinhard Diestel. How to demonstrate the duality theorem between ...
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0answers
32 views

Paths in a linear graph

Is there an expression for the number of paths of length $k$ from one end of a linear graph of length $N$ to the other?
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1answer
53 views

Infinite resistor problem from a graph theory standpoint

I am trying to understand the infinite grid of resistors problem from a graph theory stand point(classic xkcd/google problem). Since effective resistance is the same as the commute time, this is ...
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0answers
29 views
2
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2answers
32 views

Graph's Matching and edge covering

Let $G$ be a graph and $M$ a match with maximum size and $F$ an edge cover with minimal size. Prove that: $|M|+|F| = |V|$ That means that the number of all Matches with maximum size and the number of ...
1
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1answer
36 views

How many distinct directed acyclic graphs are there?

Given $|V|=4$ and $|E|=3$, how many distinct directed acyclic graphs can be formed? Isomorphic graphs should be counted as one. There is one where three periphery nodes point to a central node. ...
2
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0answers
48 views

proof of Triangle Removal Lemma

Where can I find a proof of the following version of Triangle Removal Lemma (or any version equal to it)? Let $G(V,E)$ be a graph on $n$ vertices such that it contains $\varepsilon n^3$ triangles, ...
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0answers
69 views

Integral identity graphs — smallest example

From Paulus Graphs. "The (25,2)-, (25,4)-, and (26,10)-Paulus graphs have the apparently rather unusual property of being both integral graphs (or asymmetric) and identity graphs (a graph spectrum ...
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0answers
54 views

Find the Eigenvalues of Petersen Graph

Petersen graph is k-regular graph on $n$ vertices and $m$ edges. We can find eigenvalue of $k-regular$ graph by characteristic polynomials of $G$ (denote $\chi_G (x)$) and $L(G)$ (denote $\chi_L (x)$)...
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1answer
19 views

Adjacency matrix is totally unimodular

Prove that the adjacence matrix of a simple graph is totally unimodular... I know incidence matrices are totally unimodular because in every column there is a 1 and a -1... makes things easier. Any ...
6
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0answers
137 views

A graph problem

Consider the following graph problem. We are given a set of vertices $A_i$, $B_i$, and $C_i$ where $i \in \{1,2,3 \}$. For each vertex, there is a corresponding weight where the weight of vertex $A_i$ ...
2
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1answer
27 views

Finding a recurrence for number of paths in a certain tree

I have a graph which looks like this: The question is to find a recurrence for $a_n$ - the number of paths of length $n$ that start in vertex $A$. How do you tackle these kind of problems? There is ...
3
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2answers
30 views

Metric dimesnion

Metric dimension of a graph $G$ can be defined as the minimal cardinality of the subsets $A\subseteq V(G)$ with the following property; For any two vertices $u$ and $v$, we are able to find a vertex $...
2
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0answers
33 views

Graphs derived from colorings of locally finite graphs

Let us assume we are in the following situation: We have a connected regular locally finite graph $G=(V,E)$ and let us call the degree of an arbitrary (and therefore any) vertex $d$. In addition we ...
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2answers
64 views

Finding the maximum length of a minimum spanning tree

Graph G has 4 vertices/nodes and 5 edges. It is also connected. Its edges have the following weights: 5, 8, 10, 16, 18. The minimum length for a minimum spanning tree of graph G would be ...
2
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0answers
32 views

Irregular self complementary graph.

Is there a family of irregular self complementary graph? Or are all self complementary graphs regular?
1
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1answer
47 views

What is the expected number of triangles contained in this graph?

I can't seem to understand this question and I really don't know where to start. Could someone please give an explanation as to how to go about answering this? A simple graph is formed randomly on ...
2
votes
2answers
24 views

Labelled spanning trees of $K_n-e$

Let $e$ be an edge of $K_n$- the complete graph on $n$ vertices. Prove that the number of labelled spanning trees of $K_n-e$ is $(n-2)n^{n-3}$. I think the answer lies in using some modified form of ...