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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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What is the correct statement of this Erdös-Gallai-Tuza problem generalizing Turan's triangle theorem?

In Zsolt Tuza's Unsolved combinatorial problems I, Problem 46 is the following conjecture: Let $G$ be a graph on $n$ vertices. Let $\alpha_1$ be the maximum number of edges of $G$ such that every ...
darij grinberg's user avatar
3 votes
2 answers
100 views

Maximum density of a double-node-colored graph

Let $G$ be a graph with the following properties. $C$ is a set of colors. The nodes of $G$ are $C \choose 2$. For any node $v$, the neighborhood of $v$ is rainbow -- that is, for any two nodes $x, ...
GMB's user avatar
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0 votes
1 answer
398 views

Forbidden Tripartite Graphs

I was looking at extremal graph theory. I have understood the proofs of upper bounds for the Zarankiewicz problem which basically states: What can you say about the edges of a graph with $n$ vertices ...
Halbort's user avatar
  • 1,129
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,\...
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6 votes
1 answer
796 views

No big clique minor but a big grid minor

I was wondering if the following result is known (or if there's a nice short proof without treewidth/brenchwidth related theorems): as the title says, suppose you have a graph without a big clique ...
Cosmin Pohoata's user avatar
4 votes
1 answer
1k views

Complete Bipartite Subgraph of Dense Bipartite Subgraph

Q1: Consider a $2^n$ by $2^n$ bipartite graph with at least $(1-\epsilon)2^{2n}$ edges. For any $\epsilon > 0$ and $n$ large enough, is it always possible to find a $2^{(1-f(\epsilon))n}$ by $2^{(1-...
Patrick's user avatar
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6 votes
1 answer
582 views

On Turan's theorem

Turan's theorem provides minimum number of edges of a graph on $n$ vertices to surely contain a clique of a prescribed size. This has been generalized to regular graphs. What additional ...
Turbo's user avatar
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10 votes
0 answers
222 views

Asymptotics of subgraph densities in graphons

In Pittel (1989)'s solution to a problem of Knuth (1976) on the expected number of stable matchings between $n$ men and $n$ women under uniform random preferences, it was shown that, as $n \to \infty$,...
Yufei Zhao's user avatar
13 votes
1 answer
544 views

Induced Paths of Order 4

In a graph $G=(V,E)$ of order $n$, what fraction of the $\binom{n}{4}$ $4$-subsets of $V$ can induce the path of order four? I looked at this question 30 years ago and was never able to come up with ...
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1 vote
0 answers
62 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}\...
Turbo's user avatar
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3 votes
0 answers
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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 ...
GMB's user avatar
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4 votes
1 answer
437 views

Extremal functions for tournaments

We are looking at directed graphs with no loops or parallel edges, but given two vertices $x$ and $y$, we allow the presence of both the edge $(x, y)$ and $(y, x)$. Thus, if $G$ is a directed graph ...
Paul Wollan's user avatar
2 votes
3 answers
708 views

Non-isomorphic graphs with the same numbers of closed walks

Can somebody help me to construct two family of finite simple connected graph $G_i$ and $H_i$, $i=1, 2, \cdots,n$ ($n$ possibly large), such that: $1)$ $G_i‎\ncong H_i$ for $i=1, 2, \cdots, n$ $2)$ $...
Shahrooz's user avatar
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4 votes
0 answers
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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 ...
karpasi's user avatar
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2 votes
2 answers
1k views

Random graphs require O(n log(n)) edges until they are almost certainly fully connected - what are more concrete boundaries ?

As far as I understand my literature, the probability of a random graph being fully connected tends toward one as the number of edges approaches a value of size $O(n \log(n))$ The way I read this, ...
tdullien's user avatar
8 votes
1 answer
338 views

Spectral lower bounds on the diameter of a graph

There is such a bound, due to Mohar and McKay, using the second-smallest eigenvalue of the Laplacian $\lambda_{2}$: $$Diam \geq \lceil \frac{4}{n\lambda_{2}} \rceil.$$ This bound is very elegant but ...
Felix Goldberg's user avatar
6 votes
1 answer
243 views

Hamiltonicity criteria for sparse graphs

Given a sparse graph, how can one go about proving that it is Hamiltonian? (Assuming it actually is, of course). There are three main classes of criteria for Hamiltonicity that I am aware of: Dirac-...
Felix Goldberg's user avatar
6 votes
0 answers
76 views

Cage graphs and even cycles

Let $G$ be a $(\nu,g)$-cage graph of degree $\nu$ with girth $g$ and $n=n(\nu,g)$ vertices. Based on the known examples, I am wondering if the following can be proved/disproved: Is it true that ...
Wolfgang's user avatar
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-1 votes
1 answer
199 views

Maximum degree and matching number

Let $G=(V,E)$ be a finite graph. We write $\nu(G)$ for the matching number of $G$. Is there $\varepsilon > 0$ such that we have $$\frac{\nu(G)+\Delta(G)}{V(G)} \geq \varepsilon$$ for all finite ...
Dominic van der Zypen's user avatar
5 votes
3 answers
1k views

Erdős–Stone theorem type edge density estimates for bipartite graphs?

The Erdős–Stone theorem theory says that the densest graph not containing a graph H (which has chromatic number r) has number of edges equal to $(r-2)/(r-1) {n \choose 2}$ asymptotically. However, ...
Rune's user avatar
  • 2,416
2 votes
0 answers
44 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\...
Lwins's user avatar
  • 1,551
6 votes
3 answers
1k views

Online Library of Unlabeled Connected Graphs on n Vertices

Does anyone know of the link to an online library of of unlabeled, connected graphs on n vertices? I remember looking at such an archive a few years ago while at a Macaulay 2 workshop, but I've been ...
Gwyn Whieldon's user avatar
4 votes
3 answers
241 views

Almost all graphs have a subgraph from a large class of graphs with constant order

I will pose the question in relation to trees but the more general question that can be deduced from the title of this post is also very interesting. I suspect the question might have a very trivial ...
Jernej's user avatar
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7 votes
1 answer
560 views

Graphs with many triangles but few complete graphs on 4 vertices

Let $G$ be a graph on $n$ vertices with $an^2$ edges containing at most $an^2/2$ copies of $K_4$. If there are cubically many triangles, say $cn^3$, then there is at least one edge that is not ...
Thomas Kalinowski's user avatar
1 vote
0 answers
54 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 ...
Wolfgang's user avatar
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7 votes
2 answers
756 views

Is the feedback vertex number bounded by the maximum number of leaves in a spanning tree?

I have a graph-theoretical conjecture which I think would have been studied before, but for which I cannot find anything in the literature. Let G be a finite, simple, connected graph. Let the ...
Bart Jansen's user avatar
2 votes
0 answers
301 views

The Turán problem for graphs with given chromatic number

The ordinary Turán problem for graphs asks, "Given a graph $H$, if $G$ is an $H$-free graph on $n$ vertices, what is the largest number of edges that $G$ can have?" As is well known, if $\chi(H) = r +...
Andrew Uzzell's user avatar
2 votes
0 answers
134 views

Even cycle constrained edge coloring

Is minimum colors needed to assign colors to edges of complete graph $K_n$ so that every $2t$ simple cycle where $t\in\Big\{1,\dots,2\Big\lfloor\frac{n}2\Big\rfloor\Big\}$ contains atleast $t+1$ ...
Turbo's user avatar
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6 votes
0 answers
315 views

Algorithms for computing the Resilience of Graphs

The definition of resilience with a graph $G$ w.r.t to a monotone property $\mathcal{P}$ is well known. (Global resilience) Let $\mathcal{P}$ be an increasing monotone property. The global ...
Pavan Sangha's user avatar
1 vote
0 answers
131 views

What are constructions for induced $C_5$-free graphs?

During a recent workshop, the question came up whether there are some constructions for graphs that are induced $C_5$-free, but they contain "everything else," so we don't want to forbid $C_5$'s, ...
domotorp's user avatar
  • 19.1k
5 votes
1 answer
830 views

Connected components of large induced subgraphs of hypercubes

Let $H$ be the $n$-dimensional hypercube, i.e. $\{0,1\}^n$ with edges between two vertices if and only if they differ in exactly one co-ordinate. We say that an edge is in direction $i$ if its ...
Sukhada Fadnavis's user avatar
5 votes
1 answer
1k views

Characterization of infinite paths in graphs

First an introduction. A directed graph we all know what is, and a graph is serial whenever every vertex has a successor. I do not consider the empty graph. A pair $(\mathcal{G},s)$ is called a ...
Pål GD's user avatar
  • 126
2 votes
2 answers
721 views

Minimal Non-planar Extensions of a Graph

Given a planar graph $G=(V,E)$ with vertices $V$ and edges $E$, call $\bar G = (V,\bar E)$ a non-planar extension of $G$ if $\bar G$ is non-planar and $E \subset \bar E$. I'm interested in minimal ...
Ryan Thorngren's user avatar
9 votes
0 answers
442 views

An extremal problem for graphs having every edge contained in a 4-clique

This is a follow-up to Graphs with many triangles but few complete graphs on 4 vertices I'm looking for an upper bound for the difference between the number of edges and the number of 4-cliques in a ...
Thomas Kalinowski's user avatar
2 votes
1 answer
482 views

Suppose the independent number of a graph is bounded. How small the clique number can be?

Suppose the independent number of a graph is bounded. How small the clique number can be? linear? It seems to be a natural problem to ask. but I could not find any reference. Thanks.
Jian Li's user avatar
  • 23
8 votes
1 answer
328 views

Maximal class of simple graphs of order $n$ with mutually distinct numbers of spanning trees

This problem in some ways related to this post. Let $A_n$ be the set of all integers $x$ such that there exist a connected simple graph of order $n$ having precisely $x$ spanning trees. Study the ...
Jernej's user avatar
  • 3,463
11 votes
0 answers
310 views

How many n/2-cycles can a cubic graph have

Given a simple cubic graph with $n$ vertices (which implies that $n$ is even), what is a good upper bound on the number of cycles of length $n/2$ it can have? A random cubic graph has $\Theta((4/3)^n/...
Brendan McKay's user avatar
9 votes
1 answer
579 views

What is the state of the art for the Turán number of $K_{4,4}$?

In Chung and Graham's "Erdős on Graphs: His legacy of unsolved problems," they discuss several open problems concerning Turán numbers for bipartite graphs. There is a construction which gives graphs ...
Matthew Kahle's user avatar
1 vote
1 answer
175 views

Distribution of Induced Subgraphs of Extremal Ramsey Graphs

Choose $k$. Let $G = (V,E)$ be a graph on $n = R(k,k)-1$ vertices (that is, $G$ is an extremal example for $R(k,k)$, and $g : E \to \{r, b\}$ be an edge 2-coloring such that there is no monochromatic $...
fkenter's user avatar
  • 316
2 votes
1 answer
250 views

Is there a polynomial upper bound for number of holes over following class of graphs?

A hole is chordless cycle that length of the cycle is four or more. In this post I asked: What is the maximum number of holes that a simple graph on n vertices can have? Gil Kalai answered that ...
j.s.'s user avatar
  • 519
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 $M_{(k,e)}$ ...
Michael's user avatar
  • 267
1 vote
2 answers
623 views

Bounds on the independence number of a graph

If $G$ is a graph with $n$ vertices and $\frac{nk}{2}$ edges, $k\ge -1,$ then $a(G)\ge \frac{n}{k+1}$. Why? (Here $a(G)$ is the independence number).
Arash 's user avatar
  • 11
2 votes
0 answers
78 views

Maximum cardinality general factor of a graph

Given a graph $G=(V,E)$ and a set of integers $B(v)$ associated to each vertex, a general factor of $G$ is a set of edges $F\subseteq E$ such that the degree of each vertex $v\in V$ in the graph $(V, ...
George Octavian Rabanca's user avatar
2 votes
0 answers
285 views

Bipartite independence number

Consider a balanced bipartite graph $G=(U,V,E)$, i.e., a bipartite graph with $|U|=|V|$. An independent set $I$ of $G$ is balanced if $|I \cap U| = |I \cap V|$. The bipartite independence number of $...
alezok's user avatar
  • 418
3 votes
1 answer
334 views

Degree conditions for k-factor

I am looking for a simple degree conditon that ensures the existence of a k-factor in a graph. The k is supposed to be relatively high and I don't mind the condition being a bit strict. Ideally, ...
Felix Goldberg's user avatar
1 vote
1 answer
180 views

Have you come across this kind of "degree" concept?

Let $A \subseteq V(G)$ be a set of vertices in a graph $G$ and let $v \in V(G)$ be some vertex. Define $d_{A}(v)$ as the number of neighbours of $v$ inside $A$. Now suppose you have a graph whose ...
Felix Goldberg's user avatar
4 votes
0 answers
434 views

Smallest matrix covered by many random n by n matrices

We say that a matrix $M$ can be covered by a (smaller) matrix $N$ if every entry in $M$ is contained in some submatrix of $M$ that exactly equals to $N$, up to reordering the rows and columns of $N$. ...
Trinh Huynh's user avatar
1 vote
0 answers
324 views

graphs with maximal number of paths of given length

Hi, For a given number of edges, the non directed graph which maximises the number of paths of length 2 is the quasi-star or the quasi-complete graph. Does anyone know : 1- what is the non directed ...
user20638's user avatar
6 votes
0 answers
889 views

Cliques of hyperedges

Suppose we have a graph, with multiple edges allowed. An edge-clique is a set $C$ of edges so that every two edges in $C$ share at least one endpoint. Note that any edge-clique falls into one of two ...
Dave Pritchard's user avatar
1 vote
1 answer
319 views

Graphs far from being a collection of bicliques

A biclique is a complete bipartite graph. A graph is a "biclique collection" if it can be decomposed into the disjoint union of bicliques. Denote the set of such graphs by $\mathcal{BCC}$. Given a ...
Jeremy H's user avatar
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