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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1
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1answer
36 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 ...
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 ...
0
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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
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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 ...
1
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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 $K_{3}$? ...
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
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1answer
40 views

Neighbourhood set in Graph theory [closed]

Let $G$ be any connected graph with $\Delta(G)$ be maximum degree. If $D \subseteq V(G)$ then how can we say that $\left | \bigcup \limits_{v \in D} N(v) \right | \leq |D| \Delta(G)$.
3
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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 ...
0
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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
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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
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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 ...
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
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
49 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 ...
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 ...
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
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
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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 ...
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 ...
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. $\...
1
vote
1answer
42 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; ...
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
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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, ...
0
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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 ...
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?
1
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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 ...
2
votes
2answers
33 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
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. ...
3
votes
1answer
65 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, ...
3
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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 ...
0
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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 $\chi_L (x)$)...
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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
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 $A_i$ ...
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 ...
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 $...
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 ...
0
votes
2answers
71 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
votes
0answers
32 views

Irregular self complementary graph.

Is there a family of irregular self complementary graph? Or are all self complementary graphs regular?
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
2answers
25 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 ...
0
votes
0answers
14 views

Deduce Max flow min cut from Menger's theorem

I want to deduce the max flow min cut theorem from Menger's theorem, both on arc-connectivity in digraphs. Given a network with integer capacities c, one may replace each arc a by c(a) parallel arcs ...
5
votes
2answers
39 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
1
vote
1answer
78 views

If $G$ is a graph of order $n$ such that $\delta (G) ≥ (n-1)/2$ , then $\lambda(G) = \delta(G)$

Prove that if G is a graph of order n such that δ(G) ≥ (n-1)/2 , then λ(G) = δ(G). where δ(G)= minimum degree of the graph G λ(G)= minimum edge cuts to disconnect graph G κ(G)= minimum ...
1
vote
1answer
22 views

Eulerian path in directed graphs

I am a math student and am having trouble with the following problem. It would be nice if someone hand me some solution or at least some hint. Show that in a connected directed graph where every ...
0
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0answers
34 views

Probability mass function meeting the expectation of distributions of independent Bernoulli variables

Suppose there are $n$ objects that have probabilities $p_1, p_2, \ldots, p_n$ of being selected, respectively. The sum of these probabilities is not necessarily $1$. Also assume that the first $k < ...
0
votes
2answers
27 views

Calculate order of a graph from size of graph and size if its complementary.

Given the order of a graph (without loops) n, which size is 56. And its complementary graph which size is 80. How to find out the value of n?
3
votes
1answer
39 views

Uniqueness of graph neighbourhood sizes

I was thinking about graphs the other day, and had the following questions which I suppose fall under the topic of graph reconstruction. I am not very familiar with the literature, so in case this ...
0
votes
0answers
15 views

Corona of two graphs and Cartesian product of two graphs

I would like to know whether corona of two graphs is defined for two graphs with disjoint vertex set?? Also I would like to know whether Products of of two graphs (Cartesian, tensor,strong, ...
0
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0answers
23 views

Implicitization Problem on Graphs?

I learnt the implicitization problem for varieties in introduction course on Algebraic Geometry. I am trying to understand how to formulate a similar implicitization problem on graphs where the ...
1
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0answers
24 views

Arc-bases and Point-bases: when are they different for finite digraphs?

Definitions Given a digraph $D=(X,U)$, a point-basis of a digraph $D$ is a minimal vertex set from which a dipath exist to every vertex in $D$. An arc-basis is a minimal arc-reaching set. A subset ...