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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3
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1answer
48 views

planarity of trivalent graphs with a cyclic ordering on the edges in each vertex

Let $G$ be an (undirected) trivalent graph. For each vertex $v$ of $G$ we choose a cyclic ordering on the edges coming into $v$ (so if vertex $A$ has neighbors $B, C$ and $D$ we decide whether the ...
1
vote
0answers
20 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 ...
0
votes
2answers
35 views

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

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
vote
1answer
29 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. ...
4
votes
1answer
78 views

Graph theory application of homology

I am struggling with the idea of local homology groups and would like to see an example of how to go about finding them in general. I'm thinking of the most trivial case to apply the theory of local ...
1
vote
1answer
31 views

How to find List Chromatic Number of planar graphs [closed]

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 $K_{3}$? ...
5
votes
3answers
36 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 ...
6
votes
0answers
151 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 $A_i$ ...
0
votes
3answers
30 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 ...
0
votes
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 ...
3
votes
1answer
36 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 ...
2
votes
2answers
1k views

counting simple, connected graphs

I've been thinking about this for a few days, but I haven't found a general solution yet. How many distinct simple, connected, undirected graphs are there of n labelled vertices? For example, there is ...
0
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0answers
28 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 ...
2
votes
1answer
151 views

ER graphs, expected number of triangles incident to one vertex

I'm really sorry for this question. I'm new to a graph theory, and I hope you will help me to understand one statement. Consider $ER(n,p)$ graph with $n \geq 3$ and $p \in [0,1]$. The statement ...
0
votes
0answers
23 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},{...
1
vote
1answer
72 views

Complete Graph with odd degree

It is known that the Complete Graph $K_n$ has $n^{n-2}$ spanning trees. The $K_{10}$ has $10^8$ spanning Trees. Now my question: How can I compute the number of spanning Trees with odd degree of its ...
1
vote
0answers
55 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
votes
1answer
24 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 ...
0
votes
0answers
30 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 ...
0
votes
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 ...
0
votes
2answers
49 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?
0
votes
0answers
12 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
vote
1answer
31 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 ...
0
votes
0answers
18 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 ...
0
votes
0answers
18 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 ...
4
votes
2answers
2k views

A cardinality of a graph

If I have graph $G=(V,E)\\$ What is the meaning of $|G|$? (The cardinality of G). I'd like to few words about it... Thank you!
1
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0answers
39 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. $\...
4
votes
1answer
94 views

Probabilistic method: vertex disjoint cycles in digraphs

Let us say that a di-graph is $k$-regular if every vertex has precisely $k$ out-edges. The following theorem appears in a book I am currently studying Theorem. Every $k$-regular graph $D$ has a ...
0
votes
1answer
781 views

Dijkstra Algorithm proof

I was studying the proof of correctness of the Dijkstra's algorithm . In the above slide , $d(u)$ is the shortest path length to explored $u$ and $$\pi(v) = \min_{ e\ =\ u,v:u \in S}d(u) + l_e$$ and $...
0
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0answers
13 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 ...
0
votes
2answers
317 views

When a 0-1-matrix with exactly two 1’s on each column and on each row is non-degenerated? [1]

Let $A$ be an $n\times n$ matrix with entries in the set $\{0,1\}$ which has exactly two ones in each column and two ones in each row. Give necessary and sufficient conditions for the rank of $A$ to ...
7
votes
2answers
2k views

Proving every tree has at most one perfect matching

In trying to prove that every tree, T, has at most one perfect matching, I came across this idea: Since the matchings are perfect, each vertex has degree $0$ or $2$ in the symmetric difference, so ...
4
votes
2answers
64 views

Existence of $K_3$ in a graph with $(n^2+1)$ edges

I was working on this problem for quite a long time and was unable to solve it. Any help will be appreciated. Let $G$ be a graph with $2n$ vertices ($n \in \mathbb{N})$ and $n^2+1$ edges. Show that ...
1
vote
1answer
40 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. ...
0
votes
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 ...
0
votes
0answers
17 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, ...
4
votes
1answer
70 views

Consequences of cycle space cut space duality

The cycle space cut space duality theorem for planar graphs states that: The cycle space of a planar graph is the cut space of its dual graph, and vice versa. I wish to know any consequences and ...
1
vote
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 ...
0
votes
0answers
34 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?
4
votes
2answers
44 views

Hall's marriage thereom with max-flow-min-cut

I heard that Hall's marriage theorem can be proved by the max-flow-min-cut theorem. Could you outline how that is possible? Hall's theorem says that in a bipartite graph there exists a complete ...
3
votes
0answers
98 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 ...
3
votes
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 $...
1
vote
2answers
30 views

Prove a cube graph has no even walks?

The following question was in my exam, and I didn't even have any idea on how to start, so I'm quite curious to see a proof. I was given a cube graph (the one on the left): The question was as ...
0
votes
1answer
58 views

Non-trivial graph automorphism groups with $D_n$ as subgroup

I understand that the automorphism group of an $n$ cycle graph is the dihedral group $D_n$ of order $2 n$. From the comment of @Christian, I also understand that $S_n$ is the automorphism group of the ...
-1
votes
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 ...
5
votes
2answers
40 views

L(G) is isomorphic to G iff G is a cycle

The converse is pretty obvious. If G is a cycle, then it is isomorphic to it's line graph. How to prove that if L(G) is isomorphic to G, then G is a cycle...? P.S.- Assume G is connected
0
votes
0answers
56 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 $\chi_L (x)$)...
1
vote
1answer
48 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
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 ...
2
votes
0answers
35 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 ...