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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How to read Spectral Theory of Graphs

My background is a course is Linear Algebra -Hoffman,Kunze Graph Theory-Clark. I am reading Spectral Theory of Graphs. My professor has asked me to start from the book Spectra of Graphs by ...
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15 views

To show the Petersen graph has $2000$ spanning trees.

I want to show that the Petersen graph has $2000$ spanning trees. How can I achieve it?
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5 views

Eigenvalue ratio evolution of Laplacian matrix when add edges

Consider an connected digraph, we use the classic definition of the Laplacian matrix $L$: $L=D-A$, where $D$ is the degree matrix and $A$ is the adjacency matrix. There has been many researches on ...
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An efficient algorithm to decide if a directed graph is unilaterally connected

I have been doing practise problems in designing algorithms and came across the following in a past test from an American university (see attached): A directed graph is unilaterally connected if, ...
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Counting subgraphs of bounded extremal degrees

Let $m\leq n-1$. Is there a closed expression counting the subgraphs of minimum degree $\geq m$ (resp. maximum degree $\geq m$) on $n$ labelled vertices?
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8 views

Use Gröbner bases to count the 3-edge colorings of planar cubic graphs…

I found a nice introduction on how to Use Gröbner bases to construct the colorings of a finite graph. Now my graphs $G=(V,E)$ are the line graphs planar cubic graphs, so it is 4-regular. The ...
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6 views

Decomposing a graph into $N$ planar sub-graphs that can be drawn on $N$ planes.

I would like to ask you if there is a way for checking if we can decompose a specific graph into $N$ planar sub-graphs that can be drawn on $N$ planes without an edge crossing any of the planes.
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13 views

Are there useful visual representations of magmas?

In group theory we have Cayley graphs. Are there analogous or anyway useful visual representations of magma structures? I am unsure about how to construct a graph representing, for instance, a free ...
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14 views

“Chemists triple point” in percolation theory

This is a vague question asking about the existence of a mathematical object, instead of properties of a well defined one. I am sorry if this is not the correct forum. I know if you have a random ...
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1answer
27 views

Does there exist a graph with chromatic number 4 that has no triangle or square cycles?

$K_4$ is an example of a graph that requires 4 colours to be coloured but it contains triangle cycles and a square cycle too. I've tried drawing ever more complicated graphs made up of pentagons, ...
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15 views

Class of graphs with eigenvalue $1$

Are graphs with eigenvalue $1$ of multiplicity more than $1$, important one? Please guide me to any book or article discussing such graphs.
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1answer
17 views

Graphs with weighted edges and vertices

I am considering a route planning problem, which I try to model with a graph. I understand that 1. to find a shortest path in a graph, we need to know the weights on the edges. 2. as some places are ...
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7 views

The random graph quantity $S(n: K, L)$

I am going through Degree sequences of random graphs by Béla Bollobás. On page $3$ the author introduces the quantity $S(n: K, L)$ without any explanation. Could anyone please help me in ...
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1answer
19 views

Reduction to a max flow problem from a sudoku like puzzle

Given an $n$ by $n$ grid of which some of the squares are black and some are white. I'm allowed to mark some of these squares and the question is to prove whether a given grid with given black squares ...
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1answer
24 views

Discrete Math Sequences (Graph or No Graph) [on hold]

Determine if there exists a graph whose degree sequence is the one specified. Draw a graph, or explain why no graph exists. The sequence is 5,4,3,2,1,1
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21 views

Small tree containing smaller trees

Given $n$, what is the smallest number $N=N(n)$ with the property that there exists a tree on $N$ (unlabelled) vertices that contains a copy of every tree on $n$ vertices? That such $N$ must exist is ...
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1answer
25 views

Number of Labels used in reduction of Isomorphism of Labelled Graph to Graph Isomorphism

From "Lecture Notes in Computer Science" by Christoph M. Hoffmann , Assume that both $X$ and $X'$ have $n$ vertices. We plan to code the graph labels as suitable subgraphs which we attach to the ...
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37 views

2-connected graph problem (West, Introduction to Graph Theory, ex. 4.2.15)

I am struggling with this problem for hours but it seems to be easy. Here is the problem: Proof that every vertex $v$ in 2-connected graph $G$ has neighbour $u$ such that $G - v - u$ is connected. ...
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22 views

references of discrete association scheme

I tried to find a book or paper to understanding discrete association scheme but I could not get any book for that. What is the good references for that?
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1answer
20 views

Graph nomenclature for class-grouped vertices and edges

Is there a name for the subset of graph theory dealing with vertices and edges of distinct classes? For example, I could have a graph in which each vertex must be either blue, yellow or red and each ...
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1answer
35 views

Prove this simple graph is not planar.

Graph I need to show this graph is not planar. I've attempted to find $K_5$ and $K_{3,3}$ as a subgraphs but haven't been successful yet. It's possible but unlikely this graph is planar but I haven't ...
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2answers
35 views

Connecting up boxes mathematically (Puzzle)

How would you connect each black box once to each colored box without any lines overlapping, this is racking my brain so please help. Note that you can move the boxes where ever you want. Maybe ...
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1answer
18 views

How many mappings are there between these two graphs?

Let $P_{20}$ be a path of length 20 like so: $x_0$-$x_1$-$~\cdots~$-$x_{20}$ and $G$ a cycle of order 3. Allegedly there are $3 \cdot 2^{20}$ mappings $P_{20}\rightarrow G$, which I don't quite see. ...
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1answer
58 views

What kinds of transformations preserve network topology?

I have been reading a number of "network science" papers where the authors perform transformations on networks that seem to preserve the topology of those networks. By "topology", I mean a collection ...
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1answer
23 views

graphs with smallest eigenvalue at least -1

Let $G$ be an undirected simple graph and let $A$ be its adjacency matrix. It is easy to see that $A$ is neither positive semidefinite nor negative semidefinite. I would like to know if there are ...
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48 views

Is it always possible to get MC/DC coverage on an $n$-input Boolean function with $n + 1$ test cases?

In software engineering, there is a coverage metric for testing called modified condition/decision coverage, or MC/DC for short. This metric is well-known in the avionics industry due to showing up in ...
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26 views

Show that any vertex $v$ of $P$ is half-integral.

Let $G$ be an undirected graph and define $$P=\{x \in R^{V}: x(u)+x(v) \leq 1 \:\:\text{for all edges}\:\: e=uv,\:\: x \geq 0\}$$ Show that any vertex $v$ of $P$ is half integral.
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2answers
40 views

Unsolved problems in graph theory

Is there a good database of unsolved problems in graph theory?
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1answer
49 views

Colored graph isomorphism reduction to uncolored graph isomorphism

I am trying to find a polynomial time reduction from the colored graph isomorphism to the regular graph isomorphism. Doing a search on this problem, I found this article and it seems like theorem 1 is ...
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1answer
12 views

Graph Theory Proof Degree Question

Let G be a graph of order n. Prove that if deg u + deg v ≥ n - 2 for every pair u, v of nonadjacent vertices of G, then G has at most two components.
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15 views

Degrees of vertices in a circuit must be even

Let $G$ be a graph with a circuit. Let $C$ denote the subgraph of $G$ consisting of vertices and edges of the circuit. Then for every vertex in $C$, $\deg (v)$ considered in $C$ is even. I would ...
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2answers
13 views

Removing an edge from a circuit on a connected graph

Let $G = (V,E)$ be a connected graph. Suppose $e$ is an edge in a circuit of $G$. Show that the new graph $(V,E-\{e\})$ is still connected. Attempt: Let $v,w \in V$ be vertices. Then inside $G$, ...
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1answer
29 views

show that the maximum degree of the graph is 6

Let p1, p2, . . . , pn be n points in the plane such that the distance between any two points is at least one. Let G = (V, E) be the graph such that V = {p1, p2, . . . , pn} and E = {pipj | distance ...
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Does K_4 with an edge removed contain two or three cycles?

I need to answer a question about Cycle Hitting Sets. Such a set if a set that contains at least one edge from every cycles of the graph. My question is. Say we have two adjacent faces. Are there ...
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1answer
13 views

Calculating the probability of a graph being Erdos-Renyi

Given an undirected, unweighted graph with |V| = 11 and |E|= 19 and given probability p=0.5 I have to calculate the probability of the graph being generated using the Erdos-Renyi Model. I applied the ...
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1answer
63 views

The set of isomorphisms from a right coset of the automorphism group $Aut(X)$ in $S_n$.

From "Lecture Notes in Computer Science" by Christoph M. Hoffmann , on page 22- Theorem 4 Let $X$ and $X'$ be two isomorphic graphs with vertex set $V = \{1 ..... n\}$ , Then the set of ...
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1answer
11 views

Given an undirected connected simple graph with distinct edge weights.

Given an undirected connected simple graph with distinct edge weights. If we add a constant value to each edge of graph then : single source shortest path of new graph can be changed? My attempt : ...
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20 views

Bipartite graphs whose minimal cycles have length $4$

Is there some literature about bipartite graphs whose minimal cycles all have length $4$? By that I mean that any cycle in the graph with length strictly greater than four can be divided into cycles ...
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21 views

edge probability graph

I apologize if this is something very trivial, but I couldn't find an answer to it anywhere: I have a directed graph with n = 280 nodes and ...
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17 views

Small graphs containing all trees on $n$ vertices

What do those graphs look like which contain a copy of every tree on $n$ vertices and such that no proper subgraph has this property?
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1answer
37 views

an edge coloring of $k_{16}$ with no monochromatic triangle [on hold]

My plan is to show that $R(3,3,3)$ is more than 16. So, i want to prove it with graph-theory. i know i should find an edge coloring of $k_{16}$ which contains no monochromatic triangles. Can anyone ...
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13 views

Upper bound on the product of independence number and transversal for graph

I am trying to prove if $G$ is an $n$ vertex graph such that $|E(G)| \leq \alpha(G)\tau(G)$, then $|E(G)| \leq \frac{n^2}{4}$ where $\tau(G)$ is the smallest transversal in $G$. A transversal is a ...
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22 views

What is a Hungarian forest: definition

I have a doubt in the definition of the Hungarian forest. This is from the book Matching theory by Lovasz. Let $G$ be a bipartite graph with partite sets $A,B$ and let $M$ be a matching of $G$. Let ...
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10 views

Deleting vertex v from tree T leaves degree(v) components.

I don't really know how to approach this apart from: delete vertex v entails that you've deleted degree(v) edges, when you delete an edge form a tree you are left with exactly two components, .... ...
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2answers
39 views

smallest Bipartite Graph

Definition: A graph G is bipartite if its vertices can be partitioned into two sets V1 and V2 and every edge joins a vertex in V1 with a vertex in V2. Bipartite graphs can be characterized by all ...
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2answers
14 views

Each regional of a maximal planar graph is a triangle

Show that every region of a maximal planar graph is a triangle. A planar graph G is called maximal planar if the addition of any edge to G creates a nonplanar graph. Will this proof use the ...
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1answer
43 views

Prove that a one-color $K_4$ exists in a two-color $K_{18}$

An edge coloring of a graph is an assignment of colors to the edges of the graph. I have $K_{18}$ colored with blue and red and I want to show that it contains a $K_4$ colored with just one color. ...
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2answers
42 views

Can someone give a quick explanation of what this exercise wants from me?

I am having trouble understanding exercise 1.25 from the picture below. I know what order means, but the second sentence puzzles me. I have included the exercises before it in the case that 1.25 ...
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1answer
23 views

Need combinatorial formula

Let we have a forest $F_n(P)$ with $n$ nodes defined by set $P$ of all pairs $\{\text{father}, \text{son}\}$. For instance $P=\{\{1, 2\}, \{3, 4 \}, \{1, 3 \}\}$ defines a forest $F_5(P).$ Let ...
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Graphs of (un)bounded color valence

Talking about colored graphs there is a definition given for graphs with bounded color valence. This definition is as follows: A vertex-colored graph $G=(V,E)$ has bounded color valence, if there ...