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Results tagged with graph-theory
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user 2384
Questions about the branch of combinatorics called graph theory (not to be used for questions concerning the graph of a function). This tag can be further specialized via using it in combination with more specialized tags such as extremal-graph-theory, spectral-graph-theory, algebraic-graph-theory, topological-graph-theory, random-graphs, graph-colorings and several others.
23
votes
Accepted
Cubic graphs without a perfect matching and a vertex incident to three bridges
Substitute your central vertex in your graph with a 3-cycle $abc$ so that the graph stays cubic. Now subdivide each edge in this 3-cycle. So we have new vertices $u$ connected to $a$ and $b$, $v$ conn …
- 85.6k
19
votes
Accepted
Reference request: Moore graphs
Moore posed this problem to Hoffman at a conference, so it is not in print. Hoffman makes the following remark (from "Selected Papers of Alan Hoffman with Commentary", pp. 367):
After I discussed the …
- 85.6k
19
votes
What are the applications of hypergraphs?
Hypergraphs and various properties that we can prove about them are the basis of many techniques that are used in modern mathematics. I will mention Deducing the Density Hales–Jewett Theorem from an i …
18
votes
Is the Rado graph a Cayley graph? If so, what is the group like? (And other questions...)
For the question of whether the Rado graph is the Cayley graph of any group see "An essay on countable B-groups" by Cameron-Johnson. They prove
Let $G$ be a countable group which cannot be express …
- 85.6k
15
votes
Accepted
Maximal disarrangement of $n \times n$ numbers
The problem of minimizing the maximum difference of adjacent values in a labelled graph is called "bandwidth minimization". Recently there was interest in the dual problem of maximizing the minimum su …
- 85.6k
14
votes
The Matrix-Tree Theorem without the matrix
Interesting question! There is an elementary way to prove the matrix tree theorem using only combinatorial reasoning and bijections. The result I'm referring to is: the number of spanning trees in a g …
- 85.6k
14
votes
Accepted
How much linear algebra can be done with graphs?
I will try to contribute a partial answer. First I want to comment on the Lindstrom-Gessel-Viennot determinant coming from quantum mechanics stuff, in physics this is known as the Slater determinant, …
- 85.6k
13
votes
Accepted
Is there an English translation of Kuratowski's theorem on forbidden minors of planar graphs?
In case you are asking for the original paper "Sur le problème des courbes gauches en Topologie" by Kuratowski where he first proves his characterization of planar graphs, then a translation by J.Jawo …
- 85.6k
13
votes
Accepted
Does every $C_4$-free bipartite graph lies in some finite projective plane?
This is an open problem posed by Erdos in "Some old and new problems in various branches of combinatorics" (see section 6). There hasn't been any substantial progress since then. After posing the ques …
- 85.6k
12
votes
Random noncrossing chords of a circle
The article "Random recursive triangulations of the disk via fragmentation theory" discusses many properties of the model you describe. The search word is random geodesic lamination.
- 85.6k
12
votes
How many perfect matchings in a regular bipartite graph?
To elaborate on Tony's answer, for a graph $G$ with an even number of vertices and degree sequence $d_1,\dots,d_n$ the number of matchings in $G$ is at most $\prod_{i=1}^n (d_i!)^{\frac{1}{2d_i}}$, wi …
- 85.6k
11
votes
Accepted
Is the number of vertices bounded for fixed max degree and fixed diameter?
A graph of maximum degree $\leq \Delta$, and diameter $\leq d$ can have at most
$1+\Delta\sum_{i=0}^{d-1} (\Delta-1)^i$ vertices. The graphs which attain this bound are called Moore graphs.
- 85.6k
11
votes
Accepted
What is known about graphs that permit only one colouring?
It's a bit hard to give a comprehensive answer without knowing exactly what sort of properties you are after, but here is a start. Such graphs are called uniquely colorable graphs (see here and here). …
- 85.6k
11
votes
Proofs of parity results via the Handshaking lemma
Another very famous example is Sperner's lemma. Other examples are Chevalley's theorem for $p=2$, Tucker's lemma, the fact that the number of decompositions into Hamiltonian cycles is even etc. (See …
10
votes
Accepted
A variant to the Hadwiger-Nelson problem
By considering all the rational numbers on the $x$-axis we can see that we need at least countably many colors. This is also sufficient, that is the chromatic number of the rational-distances graph is …
- 85.6k