All Questions
44 questions
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) \,...
- 986
21
votes
0
answers
441
views
Straight-line drawing of regular polyhedra
Find the minimum number of straight lines needed to cover a crossing-free straight-line drawing of the icosahedron $(13\dots 15)$ and of the dodecahedron $(9\dots 10)$ (in the plane).
For example, ...
- 4,535
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 ...
- 384
7
votes
1
answer
513
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 ...
- 121
6
votes
2
answers
392
views
Coloring of a graph representing the power set
For a positive integer $n$, let $\mathcal{P}$ be the power set of $[n]$. Consider the graph $G$ with $\mathcal{P}$ as its vertex set, and, for $S_1,S_2 \in \mathcal{P}$, the edge $(S_1,S_2)$ exists ...
- 135
6
votes
1
answer
232
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
6
votes
2
answers
349
views
extremal bipartite graph
I'm facing the following question:
Given a bipartite graph $G = (L \cup R, E)$.
Let $n = |L|$, $m = |R|$, and a parameter $k \in \mathbb{N}$, $n > m > k$.
What is a minimal possible number of ...
- 161
6
votes
1
answer
612
views
Given k, what is the minimum n such that n choose n/2 is greater than k? [closed]
I'm not an expert in combinatorics, but it sometimes comes up in my research with students in computer science (which is already pretty far away from my speciality of abstract homotopy theory). I just ...
- 30.3k
5
votes
0
answers
226
views
Diameter of subset sum graph
We have a finite set $X$, a weight function $w: X\rightarrow \mathbb{Z}^+$, and constants $k\leq c\in\mathbb{N}$.
Let the weight $w(S)$ of a set $S\subseteq X$ be the sum of the weights of its ...
- 2,896
4
votes
1
answer
886
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 (...
- 43
4
votes
3
answers
780
views
Does an $(x, bx)$-biregular graph always contain a $x$-regular bipartite subgraph?
I guess a discrete-mathematics-related question is still welcome in MO since I was new to the community and learned from this amazing past post. The following claim is a simplified and abstract form ...
- 43
4
votes
1
answer
220
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 $MH(...
- 3,004
4
votes
1
answer
371
views
Looking for a counterexample to a strengthening of the union-closed sets conjecture
[Now crossposted at math.stackexchange]
Let $\mathcal{F} = \{\{x_1, x_2\} : 1 \le x_1 \lt x_2 \le n \}$, $n \ge 8$, and let $\mathcal{G} = \{G_1, \ldots, G_n\}$ be a partition of $\mathcal{F}$ in $n$ ...
- 990
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
104
views
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
1
answer
154
views
Does Sidorenko's conjecture hold when the host graph's maxdegree/mindegree is a constant?
Does the following holds?
For every bipartite graph $H$ and every graph $G$ with $\frac{\Delta(G)}{\delta(G)}\leq 2$,
$$t(H,G)\geq t(K_2, G)^{e(H)}.$$
If not sure, is this a equal question as ...
- 349
3
votes
2
answers
275
views
Ramsey-Turán density function is well defined
Define
$$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}...
- 227
3
votes
1
answer
288
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 ...
- 1,072
3
votes
1
answer
121
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 ...
- 1,034
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
2
answers
143
views
Regarding a specific Turán number of graphs
I wish to know the latest bound on the number of edges a graph of girth greater than or equal to $t$ can have.
Specifically, I heard somewhere that a graph of girth greater than or equal to $t$ can ...
- 2,089
2
votes
1
answer
160
views
Bounds for $\mathrm{ex}(n,K_{2,\dots,2}^{(r)})$
$\DeclareMathOperator\ex{ex}$We write $K_{2,\dots,2}^{(r)}$ to denote the $r$-uniform hypergraph with vertex set $\{1,2\}\times\{1,\dots,r\}$ and hyperedge set $\{(1,1),(1,2)\}\times \{(2,1),(2,2)\} \...
- 3,499
2
votes
1
answer
354
views
Extremal combinatorics on bipartite graphs
One open question in extremal graph Theory is the so-called Zarankiewicz problem
(see for instance the wikipedia page), which ask for the maximum number of edges in a bipartite graph with a fixed ...
- 1,561
2
votes
1
answer
298
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 ...
- 805
2
votes
0
answers
190
views
The drawn diagonals divide the $N\times N$ board into $K$ regions. For each $N$, determine the smallest and the largest possible values of $K$
Let $N$ be a positive integer. In each of the $N^2$ unit squares of an $N\times N$ board, one of the two diagonals is drawn. The drawn diagonals divide the $N\times N$ board into $K$ regions. For each ...
- 237
1
vote
2
answers
386
views
Lower bound for the size of a family of sets
Consider a family $\mathcal{G} = \{ A_1,B_1,\ldots,B_m \}$ of $m+1$ non-empty finite distinct sets with the following property:
$$A_1 \cap B_k = \emptyset, 1 \le k \le m$$
Let $\mathcal{F} = \{A_1 \...
- 990
1
vote
1
answer
90
views
How to get a partite minimum co-degree in a $k$-partite $k$-uniform hypergraph?
I have a $k$-partite $k$-uniform hypergraph $H$ with $V(H) = V_1 \cup\cdots\cup V_k$ (each $|V_i|=n$ for $i \in [k]$), such that the minimum vertex degree $\delta(H) \ge Cn^{k-1}$ for a constant $C$. ...
- 123
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 $...
1
vote
1
answer
75
views
Lower bound for the sum of the number of vertices of some subgraphs of a directed graph
Let $G$ be any simple weakly connected directed graph with vertices $V$, $\vert V \vert = n$. Let $V_1, \ldots, V_m$, $m = \binom{n}{k}$ be all subsets of $V$ of size $k$.
Let $C(V_i)$ be the union of ...
- 990
1
vote
2
answers
360
views
Tight bound of Turan number for K_{1,t,t}
I'm looking for a tight bound for Turan number $ex_2(n,K_{1,t,t})$, where $K_{1,t,t}$ is the complete 3-partite graph with parts of size 1, t, and t.
The motivation is that we now $ex_2(n,K_{t,t})=O(...
- 281
1
vote
2
answers
205
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 ...
- 1,389
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 ...
- 349
1
vote
0
answers
63
views
Lower bound for the minimum of the maximum frequency of an element - with restrictions
Consider a family $\mathcal{F}$ of non-empty sets, with
$n=|\mathcal{F}|$ sets, $q=\left|\cup\mathcal{F}\right|$ elements in the universe, and $q\le n/4$.
It is known that of the $\binom{n}{2}$ ways ...
- 990
0
votes
1
answer
1k
views
Vertex cover of regular graph
(1.) How small can set $S$ of vertices in any regular undirected graph $G$ on $n$ vertices with degree $\Omega(n^\alpha)$ where $\alpha\in(0,1)$ can be such that every edge in the graph is incident on ...
user76479
0
votes
0
answers
56
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,
$$
t(H, ...
- 349
0
votes
0
answers
45
views
Another version of Sidorenko's conjecture(?)
I would like to ask a question about Sidorenko's conjecture. Here is the background of my question:
Quasi-random graphs
A sequence of graphs $(G_n)$ is called quasi-random if it satisfies certain ...
- 349
0
votes
0
answers
51
views
Inverse problem of "graph limits to graphon"
A graphon is a measurable symmetric function $W: [0,1]\to [0,1].$ By Lovasz's book "Large networks and graph limits" we know for any graph sequence $G_1, G_2, \dots G_i,\dots$ there exists a ...
- 349
0
votes
0
answers
67
views
Does Sidorenko's conjecture hold when the host graph's edge density not too small?
Does the following hold?
For every bipartite graph $H$ and every graph $G$ with $e(G)\geq 0.1(v(G))^2$,
$$t(H,G)\geq t(K_2, G)^{e(H)}.$$
If not sure, is this a equal question as Sidorenko's conjecture ...
- 349
0
votes
0
answers
67
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 ...
- 349
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|\...
- 349
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 ...
- 349
0
votes
0
answers
39
views
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 ...
- 349