All Questions
Tagged with extremal-combinatorics extremal-graph-theory
23 questions with no upvoted or accepted answers
11
votes
0
answers
195
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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 ...
- 384
7
votes
0
answers
177
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Szemerédi's regularity lemma for binary operations
Szemerédi's regularity lemma is an approximate structure theorem for
all large graphs (symmetric binary relations). There are versions for
multicolored graphs and directed graphs. Is there an ...
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4
votes
0
answers
113
views
What properties do graphs avoiding large regular subgraphs have?
Fix a positive integer $r$ and real $\delta \in (0,1)$.
Let $G$ be an undirected graph on $n$ vertices. Suppose that $G$ does not contain an $r$-regular subgraph on at least $\delta n$ vertices (i.e., ...
- 557
4
votes
0
answers
241
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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, ...
4
votes
0
answers
104
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Maximal number of smallest circuits in a matroid
It is known (see here for example) that, in a simple graph of odd genus $g$ with $n$ vertices and $m$ edges, the number of cycles of lenght $g$ is at most $\frac{n(m-n+1)}{g}$.
Since this can be be ...
- 3,446
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
$$ \min_{S\subseteq ...
- 419
3
votes
0
answers
70
views
Boundary differences in two graphs
Let $\Gamma, \Xi$ be two graphs with the same set of vertices $V$ with $n$ elements. Assume $\Gamma$ is connected. Write $\Gamma\cup \Xi$ (or $\Gamma\cap \Xi$) for the graph whose set of edges is the ...
- 20.2k
3
votes
0
answers
66
views
An extremal problem in directed path systems
The following is a common rephrasing of the well-known open problem in extremal graph theory to (asymptotically) determine $ex(n, C_8)$:
What is the asymptotically maximum $L = L(n)$ such that ...
- 1,389
2
votes
0
answers
137
views
How many edges can be in an unbalanced bipartite graph of girth $>6$?
Let $G = (V, E)$ be a bipartite graph with $n, m$ nodes in its bipartition and girth (shortest cycle length) $>6$.
There is a simple counting argument called the Moore Bounds that gives
$$|E| = O\...
- 1,389
1
vote
0
answers
141
views
Specific regularity in bipartite graphs
Let $G(A,B)$ be a bipartite graph with $|A| = |B| = n$, where $n$ is sufficiently large(thus, $o(n)/n,o(n^2)/n^2\ll 1$). The edge density of $G$ is $d = \frac{e(A,B)}{n^2}$, where $e(A,B)$ denotes the ...
- 359
1
vote
0
answers
63
views
Is there any other norms besides cut norm defined on graphon?
Let $\mathcal{W}$ denote the space of all bounded symmetric measurable functions
$W : [0, 1]^2 \rightarrow \mathbb{R}.$ For any $W\in\mathcal{W}$ we say it is a kernel and define its cut norm $\lVert ...
- 359
0
votes
0
answers
57
views
Does Forcing conjecture equals to assume the host graph is regular?
Given two graphs $H$ and $G$, the homomorphism density $t(H, G)$ is defined as the proportion of mappings from the vertices of $H$ to the vertices of $G$ that preserve adjacency. Formally,
$$
t(H, ...
- 359
0
votes
0
answers
45
views
Another version of Sidorenko's conjecture(?)
I would like to ask a question about Sidorenko's conjecture. Here is the background of my question:
Quasi-random graphs
A sequence of graphs $(G_n)$ is called quasi-random if it satisfies certain ...
- 359
0
votes
0
answers
51
views
Inverse problem of "graph limits to graphon"
A graphon is a measurable symmetric function $W: [0,1]\to [0,1].$ By Lovasz's book "Large networks and graph limits" we know for any graph sequence $G_1, G_2, \dots G_i,\dots$ there exists a ...
- 359
0
votes
0
answers
67
views
Does Sidorenko's conjecture hold when the host graph's edge density not too small?
Does the following hold?
For every bipartite graph $H$ and every graph $G$ with $e(G)\geq 0.1(v(G))^2$,
$$t(H,G)\geq t(K_2, G)^{e(H)}.$$
If not sure, is this a equal question as Sidorenko's conjecture ...
- 359
0
votes
0
answers
68
views
Proving we can minimize the number of crossings by having a planar embedding of $K_{2,2}$ encircle another out of any 2 such embeddings
Say that we draw a graph in the following way: we first draw $n$ planar embeddings of $K_{2,2}$ (that is, we first draw $n$ quadrilaterals) such there are no edges which cross. Then for each of the $...
- 1
0
votes
0
answers
43
views
Locally uniformly convexity in kernels (generalized definition of graphon) with cut norm
Let $\mathcal{W}$ denote the space of all bounded symmetric measurable functions
$W : [0, 1]^2 \rightarrow \mathbb{R}.$ For any $W\in\mathcal{W}$ we say it is a kernel and define its cut norm $\lVert ...
- 359
0
votes
0
answers
52
views
Does "epsilon-regular" equal to "cut distance less than epsilon"?
Let $G$ be a bipartite graph (vertex number sufficient large) with bipartition $(U,W)$ and edge density $d$. Does these two statement equal?
$G$ is $\varepsilon$-regular, i.e. $\big|e_G(X,Y)-d|X||Y|\...
- 359
0
votes
0
answers
49
views
Property of edge-vertex transitive graphs
Recently I am reading a paper (https://arxiv.org/abs/1504.00858) with respect to edge-vertex transitive graphs. What is the property of the graph that is edge transitive and vertex transitive? I know ...
- 359
0
votes
0
answers
39
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Does this "linear-approximated" version of Graph Counting Lemma hold?
Let $0\leq d\ll\varepsilon,\frac{1}{e},\frac{1}{v}\leq 1.$ Let $G$ be a $n$-vertices graph ($n$ is sufficient large, $1/n\ll d$) and for any $A,B\subseteq V(G)$, the edge density $d(A,B)\geq d.$ Then ...
- 359
0
votes
0
answers
102
views
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 ...
- 1,561
0
votes
0
answers
153
views
A Non-trivial intersecting set system problem
Liven large enough $k\in\Bbb N$ fix $m\in\{2,3,\dots,k\}$ and fix $4k$ cardinality set $K_{4k}$.
What is the maximum $n\in\Bbb N$ such that at some $t\geq2n-1$ there are $$\mbox{ subsets }L_1,L_2,\...
user94040
0
votes
0
answers
47
views
Possible Number of Repetation of a Submatrix
Notation:
$H$ is the adjacency matrix of graph $H'$ respectively. $H_k$ is the block or sub-matrix of matrix $H$. The adjacency matrix of graph $H_k \cup H_e$ (subgraphs of $H'$) is $M_{(k,e)}$ ...
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