All Questions
Tagged with extremal-graph-theory co.combinatorics
162 questions
25
votes
3
answers
1k
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 ...
- 34.4k
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
4
answers
1k
views
A graph with few edges everywhere
Given a graph $G(V,E)$ whose edges are colored in two colors: red and blue.
Suppose the following two conditions hold:
for any $S\subseteq V$, there are at most $O(|S|)$ red edges in $G[S]$
for any $...
- 1,359
19
votes
4
answers
1k
views
Minimal graphs with a prescribed number of spanning trees
As it's 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 that I ...
- 3,463
17
votes
3
answers
464
views
Graph that minimizes the number of b/w colorings where white vertices have an odd number of black
motivated from a physical context, we are currently interested in the following graph coloring problem:
Given a connected graph $G_n$ with $n$ vertices, how many colorings exist such that all white ...
- 323
16
votes
1
answer
596
views
Spanning trees: the last darn $1/4$
Let $\Gamma$ be a connected graph. By (Kleitman-West, 1991),
if every vertex of $\Gamma$ has degree $\geq 3$, then $\Gamma$ has a spanning
tree with $\geq n/4+2$ leaves, where $n$ is the number of ...
- 20.2k
15
votes
4
answers
639
views
Sets of points containing permutations - a Ramsey-type question
The following question arised as a side-question in a geometric problem. It has a "feel" similar to problems in Ramsey-theory, but I have not found any mention of it (also I'm not very familiar with ...
- 711
13
votes
2
answers
870
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 ...
- 156k
13
votes
2
answers
597
views
Graph in which no cycle has two crossing chords
Let $G$ be a graph which does not contain a simple cycle $v_1\ldots v_k$ and two "crossing" chords $v_iv_j$ and $v_pv_q$, $i<p<j<q$. An example of such graph is a triangulation of ...
- 109k
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 ...
user22090
13
votes
0
answers
741
views
Is there a weak strong regularity lemma?
A famous strengthening of Szemerédi's regularity lemma, due to Alon, Fischer, Krivelevich and Szegedy, allows one to partition a graph into a bounded number of pieces in such a way that not only are ...
- 29k
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 ...
- 6,962
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
10
votes
1
answer
333
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 ...
- 1,091
10
votes
1
answer
526
views
Maximum number of triangles no two of which have a common edge
For $n\in N_+$, define $f(n)$ to be the maximum number of triangles in a graph $G$ with $n$ vertices, taken over all $n$-vertex graphs having the property where no two triangles have a common edge.
Do ...
- 475
10
votes
0
answers
176
views
Largest number of simple paths between two vertices
Let $G$ be a simple undirected graph, $f(v, u)$ be the number of simple paths between $u$ and $v$ in $G$, $f(G) = \max f(v, u)$ over all pairs of vertices $v, u \in G$.
A recent IOI problem utilized ...
- 5,590
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$,...
- 986
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 ...
- 7,857
9
votes
0
answers
156
views
Minimal number of colours in distinguishing colouring of biconnected graphs
A colouring of edges of a graph is distingushing if no non-identity automorphism of the graph preserves this colouring.
Problem. Is it true that each biconnected graph possesses a distinguishing ...
- 4,535
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 ...
- 2,966
8
votes
2
answers
615
views
Embedding of planar graphs
I've recently come across the following lemma.
Lemma (Valiant): A planar graph $G$ with maximum degree $4$ can be embedded in the plane using $O(|V|)$ area in such a way that its vertices are at ...
- 83
8
votes
1
answer
587
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 ...
- 33.8k
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 ...
- 6,962
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 ...
- 3,463
7
votes
4
answers
950
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 ...
- 543
7
votes
2
answers
362
views
Reference request: monochromatic paths in edge-colored complete graphs
Given $k,c \in \mathbb{N}$, let $P(k,c)$ be the minimum $n$ such that no matter how we color the edges of the complete graph $K_n$ with $c$ colors, there is always a monochromatic path of length $k$.
...
- 71
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 ...
- 2,966
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 ...
- 495
7
votes
2
answers
8k
views
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 $...
- 19.1k
7
votes
1
answer
514
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
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(...
- 71
7
votes
0
answers
177
views
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 ...
- 50.8k
6
votes
2
answers
863
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 ...
- 3,463
6
votes
2
answers
317
views
Is there a graph that is Ramsey for $P_{2n}$ but is $C_{2n+1}-$free
Write $F\to G$ to mean that for every two coloring of the edges of $F$, there exists a monochromatic copy of $G$. Nešetřil and Rödl proved that for every graph $G$, there exists a graph $F$ such that ...
- 61
6
votes
3
answers
4k
views
Kovari-Sos-Turan theorem
Let $r \leq s$ be fixed natural numbers. Then by the Kővári–Sós–Turán theorem, any graph on $n$ vertices with at least $cn^{2-\frac{1}{r}}$ edges contains a complete bipartite subgraph $K_{r,s}$ for a ...
- 389
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\...
- 20.2k
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 $\...
- 201
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 ...
- 1,324
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
6
votes
2
answers
266
views
Lovasz local lemma for the edge model
In order to successfully apply the Lovasz local lemma, one needs the events to be relatively independent. This (sometimes) works well in the $G(n,p)$ model of random graphs, where the presence or ...
- 2,349
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-...
- 6,962
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
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 ...
- 13.4k
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 ...
- 903
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 ...
- 1,288
5
votes
1
answer
185
views
Length minimizing graphs between a finite set of points
Consider a set of $n$ points in the plane. Among all the connected graphs (trees) $T$ in the plane that have these $n$ points among their vertices, I am looking to find one such that the sum of its ...
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 $v$. ...
- 1,677
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
5
votes
2
answers
383
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 $t(\...
- 131
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 ...
