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Results tagged with graph-theory
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user 9025
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.
43
votes
Accepted
A conjecture on planar graphs
Let $L(G)=\sum_{xy\in E(G)} \min\lbrace\deg(x),\deg(y)\rbrace$.
THM. For a simple planar graph with $n$ vertices, $L(G)\le 18n-36$ for $n\ge 3$.
PROOF. Recall that a simple planar graph with $k\ge 3$ …
- 37.7k
37
votes
Accepted
How many $p$-regular graphs with $n$ vertices are there?
McKay and Wormald conjectured that the number of simple $d$-regular graphs of order $n$ is asymptotically
$$\sqrt 2 e^{1/4} (\lambda^\lambda(1-\lambda)^{1-\lambda})^{\binom n2}\binom{n-1}{d}^n,$$
whe …
- 37.7k
29
votes
Accepted
Is there a graph with 99 vertices in which every edge belong to a unique triangle and every ...
First we will prove the graph is regular.
Let $x,y$ be two non-adjacent vertices, and let $a,b$ be their common neighbours. Define $X$ to be the neighbourhood of $x$ other than $a,b$, and $Y$ to be …
- 37.7k
28
votes
Is it easy to produce hard-to-color graphs?
Since nobody seems to have addressed question 3, I will. The proofs of the 4-colour theorem are effective in the sense that they can be turned into polynomial-time algorithms. So there are no planar …
- 37.7k
19
votes
Reasons for difficulty of Graph Isomorphism and why Johnson graphs are important?
Johnson graphs do not cause difficulty to existing programs. Actually they are rather easy; nauty can handle them up to tens of millions of vertices, and so can other programs such as Traces and Bliss …
- 37.7k
16
votes
Regular graph of order 50, degree 7 and Automorphism group of order 288000. How to check if ...
There isn't any good general computational method for determining whether a permutation group has a regular subgroup. It was recently described to me by an authority on permutation group algorithms a …
- 37.7k
16
votes
Is there a Cayley graph of a non-abelian finite group that is not isomorphic to any Cayley g...
If I understand my own 1979 catalogue of small transitive graphs, this happens first at 12 vertices. The simplest example to describe (L10 in the catalogue): take the tetrahedon and cut off each of t …
- 37.7k
16
votes
What is the upper bound of $R\underbrace{(3,3,3, \ldots,3)}_\text{$k$ times}$?
All questions about Ramsey numbers for small graphs should be first checked in Staszek Radziszowki's amazing frequently updated survey. On page 40 we find the upper bound $(e-\frac16)k!+1\approx 2.55 …
- 37.7k
16
votes
Accepted
What happens to eigenvalues when edges are removed?
The smallest eigenvalue can go up or down when an edge is removed.
For "down": $G=K_n$ for $n\ge 3$.
For "up": Take $K_n$ for $n\ge 1$ and append a new vertex attached to a single vertex of the orig …
- 37.7k
15
votes
Accepted
Are all cubic graphs almost Hamiltonian?
Yes, every connected cubic graph is 3-almost-Hamiltonian.
Replace each edge by two parallel edges then follow an Eulerian circuit.
In the case of a bridgeless cubic graph, you can add a perfect match …
- 37.7k
15
votes
Which finite groups are not the automorphism group of some rooted finite tree?
If I remember correctly, the automorphism groups of trees are those groups which you can make from symmetric groups by direct products and wreath products. This is rather few groups.
An example of a …
- 37.7k
14
votes
Accepted
Genus of a graph
No. The two subgraphs can share the surface more efficiently than that. Take a graph $G$ with genus $g\ge 1$ and duplicate each edge. If you don't like double edges, subdivide them with new vertices …
- 37.7k
14
votes
Accepted
Spanning trees: the last darn $1/4$
Consider connected $G$ with $n$ vertices of degree $\ge 3$ and exactly one vertex $v$ of degree 1. Take an extra copy $G'$ of $G$ with $v'$ being its vertex of degree 1.
Now identify $v$ and $v'$ to …
- 37.7k
13
votes
Accepted
How dense is the set of asymmetric graphs?
Almost all non-asymmetric graphs have exactly one non-trivial automorphism, namely a transposition swapping two vertices. So, an accurate estimate of their number is
obtained by taking an arbitrary g …
- 37.7k
13
votes
The number of chains of chordal graphs
This is not a general answer but the big numbers don't fit properly into a comment.
f(1) = 1
f(2) = 1
f(3) = 6
f(4) = 576
f(5) = 1416960
f(6) = 120678543360
f(7) = 455010170456862720
f(8) = 9537186653 …
- 37.7k