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Results tagged with co.combinatorics
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user 20595
Enumerative combinatorics, graph theory, order theory, posets, matroids, designs and other discrete structures. It also includes algebraic, analytic and probabilistic combinatorics.
32
votes
Generalizations of the four-color theorem
The coloring of higher dimensional ball packings.
A ball packing is a collection of balls with disjoint interiors. The tangency graph of a ball packing takes the balls as vertices and connect two v …
17
votes
Accepted
Koebe–Andreev–Thurston theorem - where can I find a proof?
There are many proofs, and I'm not claiming that the following list is complete. New references are welcome.
(First proof)
Paul Koebe, Kontaktprobleme der konformen Abbildung, Ber. Verh. Sächs. Ak …
- 2,581
13
votes
Chromatic number of graphs of tangent closed balls
Update May 2016
I removed the updates in Oct 2015. I was trying to combine two copies of strongly regular ball packings to double the chromatic number. But it has been point out that my constructio …
- 2,581
11
votes
1
answer
366
views
What is known about the chromatic number for minimum-distance graphs in higher dimensions?
For a set of points in $\mathbb{R}^d$ with minimum distance $a$, the minimum-distance graph connect two points iff they are at distance $a$. We can also view it as the tangency graph for a set of uni …
- 2,581
10
votes
Accepted
Coloring of the plane
Seems to be the polychromatic number of the plane.
According to my knowledge, the value is at least 4 (due to Raiskii) and at most 6 (due to Stechkin).
See Chap. 4 and 6 of The Mathematical Coloring …
- 2,581
6
votes
Some polytopes in $\mathbb R^n$ whose vertices have coordinates 1, -1 or 0
View the two examples, I think $P(n,k)$ is the $(n-k)$-rectified $n$-hypercube or the $(k-1)$-rectified $n$-cross-polytope (same thing). I believe the notion of rectification will be very helpful for …
- 2,581
5
votes
Accepted
Stable equilibria of points on the 2-sphere
This is the famous Thomson problem. You can find a list of optimal configurations and many references on the Wikipedia page. Your intuitions for $n=7, 8, 9, 20$ are wrong, and $n=5$ is not that obvi …
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4
votes
Chromatic number of graphs of tangent closed balls
I find a paper of Hiroshi Maehara (http://link.springer.com/article/10.1007%2Fs00373-007-0702-7).
He studies packing of a) closed balls, b) balls on a table, c) unit balls, d) unit balls within a rest …
- 2,581
4
votes
1
answer
1k
views
"Codes" in which a group of words are pairwise different at a certain position
I read the following problem, claimed to be in the IMO shortlist in 1988:
A test consists of four multiple choice problems, each with three options, and the students should give an unique answer t …
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2
votes
Accepted
Block error-correcting codes over inhomogeneous alphabets
I think what you are looking for is mixed codes.
A good start point would be Brouwer--Hämäläinen--Östergård--Sloane. They are talking about mixed binary/ternary code, so for some $k$, $n_1=\cdots=n_k …
- 2,581
2
votes
Conjecture regarding closest point inside a discrete ball to a line
I think the proof of @domotrop is not complete.
I will first present my own proof. After several revisions, it is now complete. Then I will express my concern for the proof of @domotrop, and propos …
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