The extremal-graph-theory tag has no wiki summary.

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### How many simple cycles can a graph with $n$ vertices and $m$ edges have?

I am mainly interested in the smallest number of simple cycles a graph with $n$ vertices and $m$ edges must have.
For example, if $m\le n-1$, this number is $0$, then if $n\le m \le 3(n-1)/2$, it is ...

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**1**answer

60 views

### Edge-disjoint cycles in graphs

Given a graph $G=(V,E)$ and a fixed integer $k$ are there any algorithms known which would find the maximum number of edge-disjoint cycles of length $k$ in $G$? If not is there a proof that this ...

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149 views

### Complete k-partite graph covers all K_k of a graph

Suppose that we have a complete graph $G$ of $n$ vertices. What is the minimum number of complete $k$-partite graph (subgraph of $G$) that covers all the complete graph of $k$ vertices of $V(G)$? Are ...

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**1**answer

49 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 ...

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**1**answer

393 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) ...

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votes

**1**answer

216 views

### On the maximum number of $t$-subset of $\{1,\ldots, n\}$ having pairwise singleton or empty intersections

Let $n$, $t \in \mathbb N$ two natural numbers such that $0<t<n$, and let $A$ be a set of $n$ elements.
We call a quasi-partition or q-p of $A$ a subset $W \subset \mathcal P(A)$ such that we ...

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**1**answer

223 views

### Smallest Connected Graph for Given Degree Sequence

For a given integer sequence $(d_1, d_2,...,d_n)$, a natural question is if such a sequence is graphical, i.e. is a degree sequence of some graph. According to Erdős–Gallai theorem, A sequence of ...

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**3**answers

649 views

### Removal of non-isomorphic edges results in the same graph

There exists a (simple unlabeled) graph on 6 nodes with a pair of non-isomorphic edges (i.e., there is no graph automorphism that sends one edge into the other) such that removal of either of them ...

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161 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 ...

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**1**answer

205 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 ...

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**1**answer

171 views

### Can I weaken the minimum degree hypothesis in Nash-Williams' triangle decomposition conjecture?

In what follows, all graphs $G$ are $K_3$-divisible (all degrees even, number of edges a multiple of three) on $n$ vertices, where $n$ is not too small.
The famous Nash-Williams conjecture claims ...

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**5**answers

1k 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 ...

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**1**answer

218 views

### Extremal graph theory for directed graphs

In extremal graph theory, there are results such as
$$t(C_4,G)\geq t(K_2,G)^4,$$
where $G$ is an undirected graph, $C_4$ is a cycle graph on 4 nodes, $K_2$ is a complete graph of $2$ nodes, and ...

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**4**answers

333 views

### Extremal examples for a folklore lemma on subgraphs of large minimum degree

It's a well known fact that a graph $G$ of average degree $d$ has a subgraph $G'$ of minimum degree at least $d/2$ and that the constant $1/2$ cannot be improved. The proof I know, which proceeds by ...

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votes

**1**answer

100 views

### Point-Hyperplane incidence in finite projective spaces

Let $P$ be a finite projective space of order $q$ and dimension $d$. I am interested in finding the least $k$ such that for any set $S$ of $k$ points of $P$, and for any set $S'$ of $k$ hyperplanes of ...

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**1**answer

690 views

### Paul Erdős: Determine or estimate the number of maximal triangle-free graphs on n vertices

Among the collections of the open problems of Paul Erdős on the website of
Professor Fan Chung, there is one called "number of triangle-free graphs".
...

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**1**answer

136 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:
...

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432 views

### Strongly connected directed graphs with large directed diameter and small undirected diameter?

This question is an attempt to make progress on domotorp's interesting challenge. This question was originally asked in two parts; the former of which was answered by Ilya Bogdanov, and the latter of ...

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**1**answer

258 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 ...

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**1**answer

171 views

### Efficient isomorphic subgraph matching with similarity scores

I'm a computer vision PhD student, and I'm looking for an efficient approximation to the following problem, which could end up helping in image to image matching. Failing that, pointers to relevant ...

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146 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 ...

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360 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 ...

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**1**answer

472 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 ...

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178 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 ...

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115 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 ...

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387 views

### Minimal graphs of prescribed girth and chromatic number

The well known result of Erdős, states that
Given integers $g > 2$ and $k > 1$ there exist a graph $G$ with $\chi(G) \geq k$ and girth at least $g.$
What I am wondering is
When can we ...

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**1**answer

305 views

### How to find or constrain “particularly good” (two-sided) spectral expanders?

I'm new to graph theory, but a response to a question I asked a while ago introduced me to the concept of expander graphs.
A k-regular graph (henceforth "graph") on n nodes has eigenvalues k = λ1 ≥ ...

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274 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, ...

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**1**answer

295 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 ...

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**1**answer

107 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 ...

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332 views

### Maximum number of edges $f(n,k)$ in a graph on $n$ vertices with no $k$-core?

The $k$-core of a finite graph is defined as follows. Delete all vertices of degree $< k$ and repeat until there are no such vertices left. If there is a nonempty subgraph remaining, necessarily ...

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**1**answer

163 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 ...

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480 views

### 6-regular bipartite graphs with no 8-cycles

I'm looking for simple 6-regular bipartite graphs with no 8-cycles, as small as possible. It doesn't matter if there are 4-cycles or 6-cycles, provided there are no 8-cycles. Such graphs must exist ...

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257 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 ...

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**1**answer

273 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 ...

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**1**answer

138 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 ...

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**1**answer

189 views

### Help on the following extremal problem?

An induced cycle is a cycle that is an induced subgraph of G; induced cycles are also called chordless cycles or (when the length of the cycle is four or more) holes.
Can anyone please tell me what ...

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743 views

### Largest graphs of girth at least 6

Let $e_6(n)$ be the greatest number of edges in a simple graph with $n$ vertices and girth at least 6.
Let $G_6(n)$ be the set of simple graphs of order $n$ with girth at least 6 and $e_6(n)$ edges.
...

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241 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 ...

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799 views

### Minimal graphs with a prescribed number of spanning trees

As its long ago since Erdős died and mathoverflow is the second best alternative to him (for discussing personal problems), I'd like to start a fruitful discussion about the following problem ...

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**1**answer

257 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, ...

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209 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 ...

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**1**answer

443 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**answer

395 views

### 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 ...

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495 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 $i=1, 2, \cdots, n$
$2)$ ...

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**0**answers

413 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 $N$. ...

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**1**answer

580 views

### If many triangles share edges, then some edge is shared by many triangles

Let $G=(V,E)$ be a graph.
Let $t$ denote the number of triangles in the graph, and $x$ denote the number of pairs of distinct triangles that share an edge.
(For example in $K_4$ we have $t=4$ and ...

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688 views

### Number of Geodesic Paths Passing Through a Vertex in an Expander Graph

Let $\{G_{n}\}$ be a sequence of $k$-regular expander graphs. For each $n$ assume that each pair of nodes in the graph is transmitting a unit load of traffic and the traffic goes through the minimum ...

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**1**answer

388 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 ...

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**0**answers

116 views

### Smallest size for an incomplete tournament with property $S_k$

By a well-known probabilistic argument due to Erdos, if $k>1$ is an integer then for all large enough $n$, there an asymmetric relation $R$ on $X=\lbrace 1,2, \ldots ,n \rbrace$ (i.e. $R \subseteq ...