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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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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13 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.
2
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1answer
16 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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0answers
6 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 ...
1
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1answer
18 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 ...
-2
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1answer
23 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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0answers
18 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 ...
1
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1answer
21 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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0answers
35 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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0answers
21 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
19 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 ...
2
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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 ...
0
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1answer
16 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. ...
1
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1answer
56 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 ...
0
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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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0answers
45 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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0answers
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
39 views

Unsolved problems in graph theory

Is there a good database of unsolved problems in graph theory?
3
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1answer
44 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 ...
0
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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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0answers
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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0answers
19 views

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
62 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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0answers
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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0answers
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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0answers
16 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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0answers
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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0answers
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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0answers
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
38 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 ...
3
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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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0answers
34 views

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 ...
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57 views
+50

How to number the left-hand turn paths of planar bicubic graphs?

When you draw a planar cubic bipartite graph $\Gamma$ and 3-color its edges you can use this as an orientation $\mathcal O$. Definition A left-hand turn path on $(\Gamma, \mathcal O)$ is a closed ...
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1answer
22 views

Solve logistic problem with graph - fitting boxes

Suppose you have $n$ boxes, each of which falls into one of the $k$ sizes, and you want to nest smaller ones into larger ones, such that no two boxes $A$ and $B$ are nested inside the same box, if ...
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1answer
19 views

prove that k-cube has $2^k$ vertices, $k$ $2^k$ $^-$ $^1$ edges and is bipartite.

The k-cube is the graph whose vertices are the ordered k-tuples of 0's and 1's, two vertices being joined if and only if they differ in exactly one coordinate. Show that the k-cube has $2^k$ vertices, ...
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1answer
35 views

Translate this problem to graph theory

Say I have a size $k$ set called $S_k$ with elements that are natural numbers (repetitions are allowed). For instance $\{2, 8, 6, 6, 1, 3\}$ is a valid set for $k = 6.$ I am trying to find the least ...
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1answer
8 views

Size of minimum vertex cover on complete graph

I know that the maximum size of an independent vertex set, also known as the independence number and denoted by $\alpha$, plus the the size of the minimum vertex cover, $\tau$, is equal to the number ...
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35 views

In graph theory, what would a negative number of vertices mean? [closed]

I'm interested in expanding the concept of vertices into negative numbers. What properties would they have? I'm thinking perhaps extending V - E + F = 2 to begin.
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1answer
23 views

Let $G=(V,E)$ be the Rado Graph. Suppose $V_1 \cup V_2=V$. Show that one of the induced subgraphs of $G$ on $V_1$, $V_2$ is also Rado.

I started off with: one of $V_1$ or $V_2$ must be countably infinite. WLOG (without loss of generality) Let $V_1$ be countably infinite. I am not sure how to proceed after this.
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1answer
14 views

How to prove that adjacency matrix is L.I. if a graph contains no cycles?

Consider the vector space $F_n^2$ , and, as usual, let $e_i$ denote the vector with 1 at the $i$th coordinate, and 0 at all others. Call a vector of the type $e_i + e_j$ an edge vector, (think ...
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0answers
30 views

What is a “box” in this context “let $B ⊂ R^n$ be a box that can be partitioned into boxes”? [closed]

Let $B ⊂ R^n$ be a box which can be partitioned into a finite number of boxes, each of which possesses a direction in which its length is an integer. Show that $B$ also has an integer length ...