Questions tagged [graph-theory]
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.
25,008 questions
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A family of lines with more than four points is 2-colorable
Let $E$ be a finite set of points and $\mathcal{F}$ be a collection of subsets of $E$ (lines) such that
each line has cardinal at least 4
two distinct lines intersects at exactly one point.
Show ...
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The exact meaning of ‘subject to that’ in this context
In the following sentence from the paper (see Page 4, the proof of Lemma 3.4) (see the paper in https://doi.org/10.1016/j.disc.2023.113431) on extremal graphs:
Let $G$ be an edge-extremal graph in ...
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Max–min assignment on a DAG when nodes have candidate values with compatibility constraints
I have a DAG where every node has a (usually small) set of candidate integers. A candidate a is compatible with b if (a | b) or (b | a). For every root I want to choose one candidate per node to ...
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Coloring a graph with K-colors
Given a graph, I wish to try and color it with the minimum number of colors (0,1...k).
Each 2 connected vertices must have different colors.
I proposed the following algorithm - Initialize an array of ...
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Justification for Definition of Directed and Undirected Graph
It is commonly known that directed graphs are defined as a double $G_d:=(V,E)$ such that $E \subseteq V^2$, and that undirected graphs $G_u:=(V,E)$ such that $E \subseteq \left\{ \{a,b\}\Big\vert a \...
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The triangulations of one graph all come from the other two [closed]
In graph theory, a triangulation means adding edges to make a graph maximal planar. The following picture is denoted by $G$.
The following picture is $G_1$.
The following picture is $G_2$.
Now ...
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Are bounds known on the number of quadrilaterals (or 4-cycles) of the putative Conway 99 graph?
The putative Conway $99$ graph is a defined as an undirected graph with $99$ vertices, in which each two adjacent vertices have exactly one common neighbour, and in which each two non-adjacent ...
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The permutation model is an expander with high probability
In the book Computational Complexity: A Modern Approach you can find the following problem
this is actually a multi-graph with loops. Their definition of expander and a hint are attached below
My ...
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Minimum nodes in a 36 edged shape with non-crossing edges and coloured nodes, with no similarly coloured nodes touching
I've been struggling and have managed to build a couple of networks with relatively low nodes but the correct edges but then I can't colour them correctly. I noticed an observation with lower numbers ...
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What's the pattern behind the algorithm of repeatedly taking a number and subtracting its reverse digits?
The algorithm is quite simple. Let's start with some definitions.
Take an integer $k$ of length $N$ we can denote its digits from left to right as $k=k_0k_1... k_{N-1}$.
Now let $k_{rev}$ be the ...
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Upperbounds on the maximum independent set for Conway's 99 Graph (if it exists).
I am just curious about Conway's 99 Graph Problem, it asks:
Does there exist a graph where every edge belongs to a unique triangle and every non-edge belongs to a unique quadrilateral?
For purposes ...
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What is the number of connected components of this graph?
Consider natural numbers $e_1 < e_2 < ... < e_k <n$. Define the graph $G$ to have the vertex set $V= \{1,\dots, n\}$ and edge set $E = \{(a,b)\in V^2 \: |\: \exists j=1,\dots, k: a = b\pm ...
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In a simple graph $G$ with $7$ vertices $v_1,\ldots,v_7$, if $\deg v_i=i$ for $i=1,\ldots,6$, find $\deg v_7$.
Let $G$ be a simple graph with $7$ vertices $v_1,\ldots,v_7$. If $\deg v_i=i$ for $i=1,\ldots,6$, find $\deg v_7$.
Attempt: Let $\deg v_7=d$. By the handshaking lemma, we obtain $\sum\limits_{i=1}^{7}\...
- 2,767
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Are maximal subgraphs related with posets?
I read that given a graph $G$, we say $S$ is a maximal subgraph of $G$ with property $p$ if for all $S'$ contained in $G$ which strictly contain $S$, then $S'$ does not have property $p$.
I was ...
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Subset sums where I have to ensure multiple different sums.
Base problem is "What is the smallest c such that every subset of S = {1, 2, ..., 100} with c elements contains at least one pair whose sum is a perfect square?"
Using disjoint pairs {1,99},{...
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Does a simple graph really have to be $2$-connected if for every triple $(x,y,z)$ of distinct vertices, there is an $x,z$-path through $y$?
Reportedly this is problem 4.2.8 in “Introduction to graph theory” by West.
Prove that a simple graph $G$ is $2$-connected if and only if for every triple $(x,y,z)$ of distinct vertices, $G$ has an $x,...
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What is a worst case graph for this streaming algorithm?
Consider the following streaming algorithm for maximum matching in a weighted graph.
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Am I right if I claim that the graph $K_1$ is connected but not 1-connected?
$K_1$ is connected, I think, as for any two vertices $a$ and $b$ in the graph, those must both be equal to the graph's only vertex, which makes that a path (of length zero) from $a$ to $b$.
However, ...
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What is the relation between matchings and edge coverings?
Let $G$ be a graph with no isolated vertices.
A set of vertices in $G$ is called
independent if no two vertices in the set are incident on the same edge;
vertex cover if every edge is incident to ...
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Does Serre's property $FA$ imply property $F\mathbb{R}$?
In the area of geometric group theory, a group $G$ has Serre's property $FA$ if any action on a tree must have at least one fixed point.
However, this article of Culler and Vogtmann defines a property ...
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How can I practically identify the 3-manifold whose triangulation is given by the clique complex of a graph?
Context: I like local graphs (neighbourhood graph of each vertex is isomorphic).
I like to look at these graphs' clique complexes, i.e. moving from a combinatorial to geometry and topology view and vv....
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Minimum # of black squares to guarantee uniqueness of loop visiting all white squares
I believe context will help before the statement of the problem. I was asked
In the picture below, can you find a closed, non-intersecting loop visiting every white square exactly once? (If you do, ...
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Question about a specific graph theorem
I have been told to prove the theorem that a graph $G$ has an independent set of size at least $k$ containing vertex $v$ iff $G-N(v)$, where $N(v)$ is the set of vertices containing $v$ and all of its ...
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Does Birkhoff-von Neumann's Theorem easily imply Hall's Marriage Theorem or equivalent?
The following is a collection of combinatorics theorems commonly refereed as "equivalent" due to one being easily derived from another.
Theorem (Hall).
A finite bipartite graph $G = (A\...
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Is it true that a graph has a fractional perfect matching if and only if $i(G\setminus S)\leq |S| $ for all $S\subset V(G)$?
Is it true that a graph has a fractional perfect matching if and only if $i(G\setminus S)\leq |S| $ for all $S\subset V(G)$? And why?
Here $i(G\setminus S)$ denotes the number of isolated vertices of ...
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