# Questions tagged [extremal-graph-theory]

Study of graphs satisfying a property that are maximal or minimal with respect to some parameter. A classic example is Turán's Theorem, which exactly characterizes the densest graphs on $n$ vertices without a $K_t$ subgraph.

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### 4-cycles vs eigenvalue information on quasi-random graphs

My (philosophical) question arises from reading the wonderful paper of Chung-Graham-Wilson where the authors introduces the notion of quasi-random graphs.
The main purpose of the paper is to show ...

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226 views

### Graph metric approximating Euclidean metric

I've been reading Wolfram's recent articles about graph/mesh/grid structures as an analogy for physical space, and it seems to me that there will be a problem getting the notion of distance to work ...

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### Expected number of bridges in a random subgraph

I am researching connectivity in random subgraphs and have come across the following problem.
A bridge between two vertices $a$ and $b$ of a graph $G$ is an edge $e$ such that removing $e$ from $G$ ...

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260 views

### Sum of degree differences for simple graphs

For a simple graph $G$ on $n$ vertices, let us define
$$\mathcal{I}_{n}(G)=\sum_{i,j=1}^{n}|\deg\ x_{i}-\deg\ x_{j}|^{3}.$$
I know that there are many different topological indices defined and ...

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93 views

### Density of bipartite $d$-degenerate graph

A graph $G$ is $d$-degenerate if every subgraph of $G$ contains a vertex of degree at most $d$. It is known that an $n$-vertex $d$-degenerate graph has at most $d(n-1)$ edges. However, if we are given ...

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### At what aspect ratio does the Ruzsa-Szemeredi Theorem begin?

One of the many equivalent phrasings of the Ruzsa-Szemeredi theorem is as follows. Suppose one has a three-layered $n$-node graph $G = (V=L_1 \cup L_2 \cup L_3, E)$, and one can partition $E$ into ...

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67 views

### Matching book embedding of Cartesian products of graphs

In the book embedding of a graph $G$ , each vertex of $G$ is placed on the spine and each edge is placed in the pages without crossing each other edge. If vertices have degree at most one in each ...

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206 views

### Population of P people, where each person knows K others, how many people mutually know each other

If you have a population of $P$ people, where each person knows $K$ others within the population (does not have to be mutual, i.e., if I know you, you don't necessarily know me), and $1<K<P$, ...

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53 views

### A simple equality for book embedding of two graphs

A book embedding of a graph $G$ consists of placing the vertices of $G$ on a spine and assigning edges of the graph to pages so that edges in the same page do not cross each other. The page number is ...

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164 views

### Lovasz local lemma for the edge model

In order to successfully apply the Lovasz local lemma, one needs the events to be relatively independent. This (sometimes) works well in the $G(n,p)$ model of random graphs, where the presence or ...

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### How many edges can be in an unbalanced bipartite graph of girth $>6$?

Let $G = (V, E)$ be a bipartite graph with $n, m$ nodes in its bipartition and girth (shortest cycle length) $>6$.
There is a simple counting argument called the Moore Bounds that gives
$$|E| = O\...

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66 views

### Can the vertices of a planar graph of min degree 3 be covered with edges of average weight ( sum of degrees) at most 14?

Consider a planar graph where every vertex is incident to at least 3 edges, and assign to each edge a weight equal to the sum of the degrees of its endpoints.
If not, what is the smallest n so that ...

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90 views

### Faithful Orthogonality Dimension of Kneser Graphs

Let us consider the complement of the Kneser graph with parameters $n$ and $n/4$. The vertex set of our graph $K$ is the set $\binom{[n]}{n/4}$ of $n/4$-subsets of $[n]$, and two vertices are joined ...

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125 views

### Chromatic number of $C_4$-free graphs

How large can the chromatic number of an $n$-vertex $C_4$-free graph be? If the maximum degree of the graph $G$ is $\Delta$, is there a bound of the form
$\chi(G) \leq O(\Delta/\log(\Delta))$ as in ...

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267 views

### Counting the forests obtainable by removing subtrees from binary trees

Let $B_h$ be the perfect binary tree having height $h$ (i.e. the binary tree with height $h$ in which all interior nodes have two children and all leaves have the same depth or same level).
For any ...

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146 views

### Combinatorial optimization for a sequential random process on graphs

Let $G(V, E)$ be a simple graph with $|V|=n$, and let $h$ be an integer in $[n]$.
We repeat $h$-many times the following operation in a sequential fashion, where the graph may change at each round. ...

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83 views

### Properties of the collection of maximal independent sets of a graph

Let $G$ be a graph and define
$\mathscr{I}(G) = \{S \subset V(G)| S$ is a maximal indepedent set of $ G\}$
1. What is known about $\mathscr{I}(G)$?
What are some of the properties of $\mathscr{I}(G)...

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315 views

### A proper definition of connectivity for hypergraphs

For usual graphs on $n$ vertices, a edge-minimal connected graph is nothing but a spanning tree of this graph. It is well-known that any spanning tree has $n-1$ edges.
I would like to know whether ...

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104 views

### The least number of edges to add to a tree that would force a certain number of edge-disjoint cycles

Let $c(n,k)$ be the least integer such that if $G$ is a simple graph on $n$ vertices with $n + c(n,k) - 1$ edges then $G$ has $k$ edge-disjoint cycles.
Clearly, $c(n, 1) = 1$ and it not very hard to ...

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146 views

### How many graphs of order n, maximum degree k, and maximum diameter d exist?

The total number of simple undirected graphs of order $n$ is
$\sum\limits_{i = 0}^{\frac{n(n-1)}{2}}{\binom{\frac{n(n-1)}{2}}{i}} = 2^{\frac{n(n-1)}{2}}$.
What is the number of simple undirected ...

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115 views

### Length minimizing graphs between a finite set of points

Consider a set of $n$ points in the plane. Among all the connected graphs (trees) $T$ in the plane that have these $n$ points among their vertices, I am looking to find one such that the sum of its ...

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86 views

### Existence of a graph with strong restrictions

Given a maximal degree $k$ and maximal diameter $d$. We identify 3 nodes, $i$, $j$, and $v$. Can an undirected graph exist, such that:
all nodes but $v$ have full degree $k$ ($v$ having a lower ...

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138 views

### Minimal number of colours in distinguishing colouring of biconnected graphs

A colouring of edges of a graph is distingushing if no non-identity automorphism of the graph preserves this colouring.
Problem. Is it true that each biconnected graph possesses a distinguishing ...

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255 views

### Blocking $a\to b\to c$ in a DAG with bounded degrees

(This is an (easy-looking) toy question for this one.)
Question. Find the smallest $\alpha$ satisfying the following:
Let $G=(V,E)$ be a finite directed acyclic graph, where each in- and out-...

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61 views

### Lower bound construction for the extremal number of $C_{2k}$-free bipartite graph

Suppose $G(V_1 \cup V_2, E)$ is a bipartite graph with parts $|V_1|=n$ and $|V_2|=m.$ What is the best known lower bound construction for the maximum number of edges in $G$ when $G$ does not have a ...

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195 views

### Maximum number of triangles no two of which have a common edge

For $n\in N_+$, define f(n) to be that for any n-vertice graph G, if any two triangle in G don't have a common edge, then G has at most f(n) triangles.
Do we have some good estimates for f(n)?
By ...

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130 views

### Blocking directed paths on a DAG with a linear number of vertex defects

Let $G=(V,E)$ be a directed acyclic graph.
Define the set of all directed paths in $G$ by $\Gamma$.
Given a subset $W\subseteq V$, let
$\Gamma_W\subseteq \Gamma$ be the set of all paths in $\Gamma$ ...

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117 views

### Halin Graphs with Highest Number of Hamilton Cycles

Halin graphs contain a Hamilton cycle and have the interesting property, that, also in the case of arbitrary real edge weights, it is possible to report one of the shortest contained Hamilton cycles ...

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### What is the smallest number of vertices in a graph whose every orientation contains a directed straight path of length 3

For a graph $\Gamma$ and a digraph $\vec H$ we write $\Gamma\Rightarrow \vec H$ if any orientation of $\Gamma$ contains an isometric and isomorphic copy of the digraph $\vec H$. Since each graph ...

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150 views

### Extremal density of a graph without a non-backtracking $2k$-cycle

The current best bound for the maximum possible density of an $n$-node graph with girth (shortest cycle length) $>2k$ is of the form
$$ex(n \ \mid \ C_{\le 2k}) = O(n^{1 + 1/k}),$$
while the ...

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854 views

### Kovari-Sos-Turan theorem

Let $r \leq s$ be fixed natural numbers. Then by the Kővári–Sós–Turán theorem, any graph on $n$ vertices with at least $cn^{2-\frac{1}{r}}$ edges contains a complete bipartite subgraph $K_{r,s}$ for a ...

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308 views

### Smallest triangle-free graph with chromatic number 5

The Grötzsch graph is triangle-free and has chromatic number 4. At 11 vertices it is the (unique) smallest graph with these properties.
What is the smallest number of vertices needed for a triangle-...

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90 views

### Percolation in torus under threshold rule

As part of my graduate research I am currently studying the last section in the paper "Random Majority Percolation" by Balister, Bollobas et al. The paper itself is very complicated but the last two ...

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421 views

### Is there a weak strong regularity lemma?

A famous strengthening of Szemerédi's regularity lemma, due to Alon, Fischer, Krivelevich and Szegedy, allows one to partition a graph into a bounded number of pieces in such a way that not only are ...

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### Two possible generalizations of a theorem of Kotlov about the Hamming Cube

The following theorem is proved here
Let $Q_n=(V,E)$ be the Hamming graph, and let $S \subseteq V$, $|S|<2^{n-1}$. Then the induced subgraph on $V \setminus S$, $Q_n[V \setminus S]$, has a ...

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244 views

### The maximal number of copies of a graph $T$ in an $H$-free graph

Problem. Let $T,H$ be fixed graphs with $H$ being a tree, not isomorphic to a subgraph of $T$. Let $ex(n,T,H)$ be the maximal number of copies of $T$ in an $H$-free graph on $n$ vertices. Is it always ...

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174 views

### Reference Request: designing a tree of “main roads” in a graph

Let $G = (V, E)$ be an undirected finite connected graph. Let $u$ be a specified vertex of $G$. Then the sum of distances
$$
\sum_{v \in V} d_G(u,v)
$$
is defined. Now we want to decrease this value, ...

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36 views

### Estimates for the drop in the sum of all pairwise weighted distances effected by a decrease of the weight of an edge

Consider simple connected undirected graphs $G = (V, E)$ equipped with a function $w\colon E\rightarrow \{x\in \mathbb{R}\colon x\geq 0\}$.
Define a function $d_{G,w}\colon V\times V\rightarrow\...

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129 views

### Subsets of a graph, maximal w.r.t. the property of inducing a subgraph with minimum degree at least $k$

Let $G=(V,E)$ be a simple undirected graph. Define an mmd$k$s in $G$ (for 'maximal minimum degree $k$ subset') to be any subset $S$ of $V$ such that
the subgraph induced by $S$ in $G$ has minimum ...

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137 views

### Minimal size of the maximal biclique

We examine a bipartite graph with two sides $R$ and $L$, and denote by $|L|$ and $|R|$ the number of nodes in each side. We know only that each node on side $R$ is connected to $k$ nodes on side $L$, ...

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57 views

### An extremal problem in directed path systems

The following is a common rephrasing of the well-known open problem in extremal graph theory to (asymptotically) determine $ex(n, C_8)$:
What is the asymptotically maximum $L = L(n)$ such that ...

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93 views

### What is the number of independent sets in graph of this type?

Suppose we have a graph $G(V,E)$
What is the number of independent sets in graph of this type?
I have an idea to use reccurence
$$|G|=|G\backslash \{v\}|+|G\backslash n(v)|$$
where $|G|$ is the ...

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57 views

### Cut norm and biclique gap?

Given real $\pm1$ matrix $M\in\Bbb R^{n\times m}$ we have that cut-norm is given by $$\|M\|_C=\max_{\mathcal I\subseteq[n],\mathcal J\subseteq[m]}\Big|\sum_{(i,j)\in\mathcal I\times\mathcal J}M_{ij}\...

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848 views

### The maximum number of edges in an even-cycle-free graph with $n$ vertices

Problem
Given any positive integer $n$, what is the maximum number of edges in an even-cycle-free graph with $n$ vertices?
Is the above problem an unsolved problem in extremal graph theory? Are there ...

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185 views

### Graph properties that imply a bounded number of edges

Many combinatorial problems can be reduced to bounding the number of edges in a given graph with $n$ vertices. Each time I encounter such a problem, I check whether the corresponding graph has a ...

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257 views

### Concepts of criticality in graph theory

A graph $G=(V,E)$ is said to be vertex-critical if removing a vertex $v\in V$ reduces the chromatic number $\chi(\cdot)$. Edge-criticality is defined in a similar manner. Moreover, $G$ is called ...

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526 views

### What is the minimum diameter of $r$-regular, $k$-connected graphs?

Let $md_r^k(n)$ be the minimum diameter over all $r$-regular, $k$-connected graphs on at least $n$ vertices. (Let us assume $r, k \geq 2$).
Problem: Find lower and upper asymptotic bounds on $md_r^...

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378 views

### Drawing trees on small number of lines in 2D and 3D

Problem. Given a tree do we need fewer lines in 3D than in 2D in order to draw it straightline and crossing-free?
(Asked 01.10.2016 by Alexander Wolff on page 20 of Volume 1 of the Lviv Scottish Book)...

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51 views

### Constructing graphs from subsets of a minimal alphabet

From an alphabet of $N$ letters, choose $n$ pairwise distinct subsets $ v_1,\dots,v_n$ of a fixed size $k$ and define a graph on $V=\{v_1,\dots,v_n\}$, which has an edge for each pair of vertices that ...

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128 views

### Maximal number of perfect matchings that pairwise form a Hamiltonian cycle

Definition: Let $MH(n)$ be the maximal number of perfect matchings (1-regular graphs) on $n$ vertices where the union of any two perfect matchings is a Hamiltonian cycle.
Question: Is it true that $...