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Results tagged with graph-theory
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user 1492
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.
34
votes
Accepted
Algebraic proof of Five-Color Theorem using chromatic polynomials by Birkhoff and Lewis in 1946
I tried to go through Birkhoff and Lewis many years ago but it is not easy because they use different variables and the style is so different to modern proofs.
In modern terms, the key idea is that i …
- 12.7k
21
votes
Accepted
Is this graph Hamiltonian?
Here is a 9-vertex graph with degree sequence 4 4 4 4 4 3 3 3 3 that does not have a Hamilton cycle (because it is bipartite on an odd number of vertices).
Edit: Here is one line of SageMath showing …
- 12.7k
15
votes
Accepted
Is the "Moebius Stairway" Graph Already Known?
They are called quartic Möbius ladders.
They are one of the fundamental classes in Johnson & Thomas's classification of internally 4-connected graphs, and crop up in matroid theory for the same reaso …
- 12.7k
15
votes
Accepted
Could the 4-color theorem be proven by contracting snarks?
Yes, the 4-colour theorem is true if and only if every snark is non-planar (this is due to Tait).
Showing that a snark has a Petersen minor would be enough to show that it is non-planar.
- 12.7k
13
votes
Accepted
Smallest strongly regular graph whose automorphism group is not vertex transitive?
There is no smaller example.
Various places, including Andries Brouwer's list of parameters and existence for small SRGS (http://www.win.tue.nl/~aeb/graphs/srg/srgtab.html) show that there are only a …
- 12.7k
11
votes
A labelling of the vertices of the Petersen graph with integers
The minimum value is $37294$ as described by F. Barrera.
I broke the symmetry a little by identifying $9$ inequivalent triples of edges to which the primes $\{41,43,47\}$ can be assigned, wrote a con …
- 12.7k
11
votes
Accepted
Does graph asymmetry imply all eigenvalues of the graph Laplacian are simple?
This is false.
There are strongly regular graphs with trivial automorphism group; these will have many repeated eigenvalues (both for adjacency matrix and Laplacian).
You can find some examples in t …
- 12.7k
11
votes
Is there any fast implementation of four color theorem in Python?
Robertson, Sanders, Seymour and Thomas, who produced a more streamlined proof of the 4-colour theorem, also addressed the algorithmic question in the paper
https://dl.acm.org/doi/pdf/10.1145/237814.23 …
- 12.7k
10
votes
Algorithm to find all (up to isomorphism) perfect matchings of quartic plane graphs
This is a very easy problem for the sizes you are proposing... a 20-vertex quartic graph seems to only have a few hundred perfect matchings (random sample), and it takes my computer approximately 2/10 …
- 12.7k
10
votes
Accepted
What is the smallest uniquely hamiltonian graph with minimum degree at least 3?
The system encouraged me to answer my own question, although it feels a bit strange to do so.
Anyway, after a bit of thinking and a (more substantial) bit of computing, I can now safely conclude that …
- 12.7k
10
votes
Accepted
Generating 21-vertex 4-regular plane graphs with 8 faces of degree 3 and 15 faces of degree 4
Brinkmann and McKay's program plantri can generate planar quadrangulations, which are planar graphs with all faces of size 4.
If you generate these on 23 vertices and then filter to keep only those of …
- 12.7k
10
votes
Is there a bipartite graph with $\sqrt{2}$ as an eigenvalue with high multiplicity, specific...
This is a limited partial answer to the question ruling out the case where the graph is regular and has four distinct eigenvalues, so the spectrum is $\{k, (\sqrt{2})^a, (-\sqrt{2})^a, -k\}$.
van Dam' …
- 12.7k
10
votes
Accepted
Can a graph be reconstructed from its cycle lengths?
Second Answer
I'm adding this as another separate answer, rather than editing the first "answer" because otherwise anyone coming late to this discussion will end up doubly confused.
So let's try aga …
- 12.7k
9
votes
Accepted
Cheeger Numbers for 3-regular Graphs
I did this calculation a few years ago (according to the timestamps on my programs).
Here is the summary of my results for $n=18$ (total of $41301$ graphs), with each line being the number of graphs …
- 12.7k
9
votes
Accepted
What is the smallest 4-chromatic graph of girth 5?
My computer tells me that there 195291625 graphs on 20 vertices with minimum degree at least 3 and maximum degree at most 6 and girth at least 5.
Sadly none of them have chromatic number 4, I just ge …
- 12.7k