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.
253 questions
2
votes
1
answer
131
views
Turán density of hypergraphs with very few edges
As usual, for an $r$-uniform hypergraph $G$, denote by $ex_r(n,G)$ the maximum number of edges an $r$-uniform, $G$-free hypergraph on $n$ vertices can have, and let $\lim \frac{ex_r(n,G)}{\binom nr}\...
- 226
4
votes
1
answer
111
views
Are there decompositions of $K_{16}$ by certain 3-regular graphs?
This is inspired by the problem of the Hoffman-Singleton Decomposition of $K_{50}$. I wanted to look at smaller variants of this kind of problem, and so naturally I started wondering:
Can the (edges ...
- 981
4
votes
0
answers
241
views
Weight transfer proof of Turán’s theorem
Turán’s theorem, which states that a $K_{p+1}$-free graph contains at most $(1-1/p)\frac{N^2}{2}$ edges, can be proven in many different ways, as pointed out, for example in M. Aigner, G. M. Ziegler, ...
- 47.8k
1
vote
1
answer
125
views
Extremal graph theory - many copies of $K_r$ imply a copy of $r$-chromatic $H$
I know that it must be a simple consequence of the Kővári–Sós–Turán (and Erdős–Stone) theorem, but I am struggling to formulate a proof: Let $H$ be a fixed-size $r$-chromatic graph. Then there exists $...
- 32.1k
4
votes
2
answers
269
views
Intuition on inequality in proving a bound on the sum of squares of degrees of a graph
Given a simple connected graph $G$ with $n$ vertices and $m$ edges, let $d_1, ..., d_n$ denote the degrees of the vertices of the graph. In this very short paper, the author prove the inequality
$$\...
- 61.9k
11
votes
1
answer
396
views
Dense triangle-free graphs and their independent sets
Recall that a graph is triangle-free if it does not contain a copy of $K_3$. Also, for a graph $G$, $\alpha(G)$ shall denote its independence number. Lastly, we will write $o(1)$ to denote quantities ...
- 306
2
votes
1
answer
300
views
Do sparse graphs contain a single regular pair?
An easy corollary of the Szemerédi Regularity Lemma is that dense graphs contain linear sized $\varepsilon$-regular bipartite subgraphs whose density is similar to that of the parent graph. As noted ...
CommunityBot
- 1
5
votes
1
answer
107
views
Maximal graphs with a property that is invariant w.r.t. vertex removal
Let $P$ be a property of graphs such that if a graph $G$ has $P$, then any graph obtained from $G$ by removal of a vertex also has $P$.
Let $g(n)$ be the maximum size of a graph of order $n$ having $P$...
- 37.7k
5
votes
1
answer
225
views
Cover a graph with small size complete graphs
Given a complete graph with $n$ nodes, if we want to use $n$ complete subgraphs to cover the graph and ask what is the minimum possible size of each complete subgraph, the answer is $\Theta(\sqrt{n})$:...
- 53
0
votes
0
answers
55
views
Comparing spectral radius of two graphs using the entry of Perron vector
Suppose we have a graph $G$.
Let $A$ be the adjacency matrix of $G$ and $x$ be the corresponding Perron vector.
Let $x = (x_1,x_2,\cdots,x_n)^t$, where $x_i$ corresponds to the vertex $i \in V(G)$.
We ...
- 199
3
votes
0
answers
108
views
Harper's theorem on the general Hamming graph
Let $G$ be a graph, and for every $S \subseteq V$, let $N(S)$ denote the neighborhood of $S$ (i.e., the set of vertices that have neighbors in $S$). The vertex expansion of $G$ is
$$ \min_{S\subseteq ...
- 419
4
votes
1
answer
465
views
Turán's theorem for cosets of groups
Let $G$ be a finite group, $G',H$ be its subgroups and $H'=G'\cap H$. For each $g\in G$, we create a map $f_g:G'/H'\rightarrow G/H: aH'\rightarrow gaH$. It's easy to see that the map is well defined ...
- 1,462
6
votes
1
answer
234
views
Graphs without short cycles and with linear number of edges
Let $f : \mathbb{N} \rightarrow \mathbb{N}$ be a non-decreasing function and let $X_f$ be the class of graphs where every $n$-vertex graph $G$ is $(C_3, C_4, \ldots, C_{f(n)})$-free, i.e. $G$ contains ...
- 655
3
votes
0
answers
331
views
Behrend's construction vs. Triangle removal lemma
I was reading Zhao's book "Graph theory and additive combinatorics" and on page 71 I came across Remark 2.5.4 which I'd like to understand.
Theorem 2.3.1 (Triangle removal lemma) For all $\...
- 330
11
votes
0
answers
195
views
Number of triangle-free graphs with prescribed number of edges
This question is posted from StackExchange since it received no answer there.
Let $f(n, e)$ be the number of triangle-free graphs on $n$ vertices and $e$ edges. From empirical evidence, I am motivated ...
- 63.9k
1
vote
0
answers
74
views
Graph removal lemma
The graph removal lemma says that for any graph $H$ and any $\epsilon>0$, there is a $\delta>0$ such that any $n$-vertex graph which contains at most $\delta n^{v(H)}$ copies of $H$ can be made $...
- 373
2
votes
0
answers
54
views
A variant of the regularity lemma that depends on the number of vertices
Suppose $G = (U \cup V,E)$ is a bipartite graph with $n$ vertices on each side.
For sets $X \subseteq U$ and $Y \subseteq V$,
let $d(X,Y) = |(X \times Y) \cap E| / (|X||Y|)$ denote the edge density ...
- 419
4
votes
1
answer
219
views
Quasi-random vs pseudo-random graphs
My question is somehow concerning terminology on extremal graph theory.
Is there any difference concerning the notion of quasi-random graph and the notion of pseudo-random graph? My feeling is that ...
- 242
4
votes
1
answer
137
views
On the number of disjoint subsets of a large set families
Let $[n] := \{1,\dots,n\}$, for some large integer $n$, and let $\mathcal{F}$ be a family of 2-element subsets of $[n]$.
The famous Erdös-Ko-Rado (EKR) theorem says that if $|\mathcal{F}| > {n - 1 ...
- 32.1k
4
votes
1
answer
205
views
Is there any study on the bounds on the number of even cycles for planar bipartite graphs?
In 1979, Hakimi and Schmeichel [1] initiated such a study by determining the maximum number of triangles and 4-cycles possible in an $n$-vertex planar graph (see also [2] for a small correction).
[1] ...
- 32.1k
0
votes
0
answers
148
views
Szemeredi's regularity lemma for countably infinite graphs?
Consider the following version of Szemeredi's regularity lemma found in the Fox and Lovasz paper, "A tight lower bound for Szemeredi's regularity lemma", arXiv: 1403.1768v1 [math.CO] 7 Mar ...
- 63.9k
6
votes
5
answers
540
views
Existence of connected set with large edge boundary
Let $\Gamma=(V,E)$ be a finite connected graph.
Pretty standard notation. Given a set $S\subset V$, write $\Gamma|_S$ for the restriction of $\Gamma$ to $S$, i.e., the subgraph $(S,\{\{v,w\}\in E: v,w\...
- 2,671
6
votes
1
answer
483
views
Average and max. hitting time to a specific vertex
Consider simple random walks that stop when reaching a given node $x$ in an undirected, unweighted and connected graph on $n$ nodes.
Let
$H(i,x)$ denote the (expected) hitting time from $i$ to $x$, ...
- 850
0
votes
0
answers
253
views
What is the weakest subsystem of Second-order Arithmetic (or its first-order part) that proves Szemerédi's Regularity Lemma?
The question is in the title. Szemerédi's Regularity Lemma is the following (according to the Wikipedia entry):
For every $\epsilon \gt 0$ and positive integer $m$ there exists an integer $M$ such ...
- 6,132
23
votes
3
answers
3k
views
Cauchy-Schwarz proof of Sidorenko for 3-edge path (Blakley-Roy inequality)
Is there a "Cauchy-Schwarz proof" of the following inequality?
Theorem. Given $f \colon [0,1]^2 \to [0,1]$, one has
$$
\int_{[0,1]^4} f(x,y)f(z,y)f(z,w) \, dxdydzdw \geq \left(\int_{[0,1]^2} f(x,y) \,...
- 33.8k
5
votes
0
answers
156
views
$K_4$ or n vertices without triangles
For which minimal $m(n)$ any graph on $m$ vertices contains either a complete subgraph on 4 vertices $K_4$ or $n$-vertices subgraph without triangles? I know a quadratic upper bound $2n^2$, but I am ...
- 109k
3
votes
2
answers
331
views
Existence of connected component with large boundary?
Question 1. Let $\Gamma=(V,E)$ be a connected
graph with $n$ vertices, all of degree $d\geq 4$. Assume every vertex has $d$ distinct neighbors. (We can think of $d$ as being much smaller than $n$, ...
- 2,175
4
votes
0
answers
102
views
Connected sets with large boundary in a multigraph
Let $G=(V,E)$ be a connected, undirected graph. Define the boundary $\partial S$ of a set $S\subset V$ to be the set of all $v\notin S$ joined to $S$ by an edge, i.e.
$$\partial S = \{v\not \in S: \...
- 2,175
1
vote
0
answers
79
views
Partitioning of a set family that avoids small intersections
Let $\mathcal{F}$ be the family of all $k$-element subsets of $[n]$. What is the smallest $\ell$ such that we can partition $\mathcal{F}$ into $\ell$ families $F_1,\dots,F_\ell$ with the property that ...
CommunityBot
- 1
8
votes
0
answers
305
views
Maximum number of cycles on regular graphs
Let $G$ be a $d$-regular graph on $n$ vertices. I'm interested in upper bounds on the number of cycles of length $k$ that hold for any such $G$. The regime I'm interested in is:
$d$ is fixed, and $...
- 133
7
votes
0
answers
152
views
Inequality of product of discrete cosines
Let $k,a,b,c$ be odd positive integers. Consider the following inequality:
$$
\sum_{x,y \in [k]} \cos^a\bigg(\frac{2\pi}{k}\cdot x\bigg) \cdot \cos^b\bigg(\frac{2\pi}{k}\cdot y\bigg) \cdot \cos^c\bigg(...
- 6,377
1
vote
2
answers
419
views
Graphs constructed from sums of perfect matchings
Consider the following natural procedure for constructing graphs from perfect matchings in graphs with even number of vertices.
Let $V$ be the set of vertices of cardinality $|V|=2n$ and let $\mathcal{...
4
votes
1
answer
195
views
Explicit constructions of regular graphs with very sparse induced subgraphs
Let $d\ge 3$ be a constant. Is there an explicit construction of an infinite family of $d$-regular graphs such that for $G$ in this family with $n$ vertices, every subgraph $H$ of on at most $\alpha n$...
- 81
10
votes
2
answers
497
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 ...
1
vote
0
answers
22
views
Topology of densest graphs whose optimal $3D$-matching can be calculated efficiently
let $A=\lbrace a_1,\,\dots,\,a_k\rbrace $ and $B=\lbrace b_1,\,\dots,\,b_{2k}\rbrace,\ A\cap B=\emptyset$ be be a partition of a graph's vertex set $V$, i.e. $V\,=\,A\cup B$.
Question:
has $G:=\...
- 13.2k
2
votes
1
answer
210
views
3-uniform tetrahedron-free hypergraph on seven vertices
My problem concerns 3-uniform hypergraphs. Let $f(n)$ be the maximal number of edges in a 3-uniform hypergraph such that no four edges form a "tetrahedron", i.e., four edges that join the ...
- 2,811
4
votes
0
answers
257
views
Graphs with high girth and low diameter
As the title says, I'm interested in graphs with high girth and low diameter.
Given a class $\Gamma$ of finite $k$-regular graphs, call a $\Gamma$-graph GD-extremal if every $\Gamma$-graph either has ...
- 3,612
2
votes
0
answers
36
views
Is homomorphism density of partially labeled graph continuous with respect to cut metric
Let $F=(V, E)$ be a finite simple graph on $n$ vertices with two labelled vertices, say $x, y$. Let $W:[0, 1]^2\to [-1, 1]$ be symmetric function. Lov'asz's book (Large Networks and Graph Limits) ...
- 371
3
votes
2
answers
276
views
Ramsey-Turán density function is well defined
Define
$$RT(n,K_l,f(n))=ex_l(n,f(n))=\max_G\{e(G): K_l \not\subset G, v(G)=n, \alpha(G)\leq f(n)\}$$
and the Ramsey-Turán density function $f_l:(0,1] \to \mathbb{R}$ as
$$f_l(\alpha)=\lim_{n\to \infty}...
- 211
1
vote
1
answer
165
views
Combinatorial graph optimization problem on integer adjacency matrices
We are given a $n\times n$ symmetric matrix $M$ whose entries are positive integers.
Let $z_{i,j}:=\frac{M_{i,j}}{M_{i,j}+\sum_{k\neq i,j}\min(M_{i,k},M_{k,j})}$ for all $1\le i<j \le n$, and
$z:=\...
- 7,226
5
votes
1
answer
171
views
Graph combinatorial optimization problem
Let $G$ be a simple graph with vertex set $V$, such that for any two vertices $u,v\in V$, we have at least $k$ edge-disjoint paths of length $2$ (i.e., formed by $2$ edges) connecting $u$ with $v$. ...
- 32.1k
2
votes
1
answer
237
views
The lower bound of number of vertices covered by maximum matching in $3$-regular graph
Let $G$ be a $3$-regular graph (cubic graph) with order $n$.
From here, the lower bound of # of vertices covered by maximum matching in $G$ is $\frac{3}{4}n$.
And from here, the lower bound is $\frac{...
- 32.1k
11
votes
5
answers
2k
views
Are all almost regular graphs obvious?
Let the maximum and minimum degress of a graph be denoted (as usual) by $\Delta$ and $\delta$ respectively.
A graph is almost regular if $\Delta-\delta=1$.
Now, here is a simple way to generate ...
- 32.1k
0
votes
1
answer
143
views
A general Turan-like question
Thinking of an edge as of a $2$-clique, it's natural to consider a slightly more general question than Turan considered in his celebrated theorem: given $r \le k \le n$, what is the maximal possible ...
- 7,836
2
votes
0
answers
64
views
Two conjectures on a special type of graph
I want to analyze a special type of weighted directed graph where out degrees is equal to in degrees at each vertex.
$n \in Z$ and $n \geq 3$ ,
Set
$x_n=\left(1,1,\dots,1\right)^T \in \mathbb{R}^{n}...
- 89
0
votes
0
answers
23
views
Does a connected $F_k$-free graph of order $n$ with the maximum singless Laplacian spectral radius belong to $Ex(n,F_k)$?
Let $G$ be a connected $F_k$-free graph of order $n$ with the maximum singless Laplacian spectral radius. Is $G\in Ex(n,F_k)$?
Here, $Ex(n,H)$ denotes the set of $H$-free graphs of order $n$ with $ex(...
- 1,182
2
votes
0
answers
61
views
Beurling’s extremality criterion for curves: two versions
I see Beurling’s extremality criterion at two places: the proof is almost identical, but the statement is very different. Below,
$$
\ell_\rho (\gamma) = \int_\gamma \rho(z) |dz|.
$$
"Extremal&...
- 1,755
4
votes
1
answer
887
views
Existence of triangle-free graphs for regular graphs of degree at most n/2
It is known that for triangle-free graphs, if they are $d$-regular, then $2d\leq n$, where $n$ is the number of vertices. In words, the degree is less than or equal to half the number of vertices (...
- 32.1k
6
votes
3
answers
443
views
Number of trees with the same matching number
Let $\sigma(n,m)$ be the number of trees with $n$ vertices $\{ v_1, \dots, v_n \}$ such that the matching number (the size of a maximum matching) is $m$.
I have been trying to compute the value of $\...
- 5,492
1
vote
1
answer
139
views
Graph with two edge-disjoint Hamiltonian paths between the same vertex-pair
Provided existence, what is the smallest graph $G(V,E)$ with two edge-disjoint Hamiltonian paths between $u$ and $v;\ \lbrace u,v\rbrace\subset V$?
- 13.2k