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Results tagged with graph-theory
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user 20595
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.
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 …
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4
votes
0
answers
169
views
Graph drawing maximizing the volume of the convex hull
Given a graph $G=(V,E)$ and a length function $\ell:E\to\mathbb{R}_+$.
An embedding of the graph into the $d$-dimensional Euclidean space is a map $f:V\to\mathbb{R}^d$ such that $||f(u)-f(v)||=\ell(uv …
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2
votes
Graphs with dangling edges
Such graphs were used in differential geometry to describe the gluing construction. See
Kapouleas, Nicolaos, Complete constant mean curvature surfaces in Euclidean three-space, Ann. Math. (2) 131, No …
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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 …
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7
votes
2
answers
178
views
Graph embedding that locally minimizes total edge lengths
I consider a graph $G$ (possibly infinite, but locally finite) embedded in the Euclidean plane $\mathbb{E}^2 \cup \{\infty\}$ such that each local perturbation of the embedding "increases the total le …
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1
vote
Graph embedding that locally minimizes total edge lengths
Finally, I find that the works of Ivanov and Tuzhilin seem to be very close to what I'm looking for.
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6
votes
1
answer
570
views
How does the distribution of Erdős number evolve over time ? How to build a model to fit the...
Let $E(n,t)$ be the number of mathematicians with finite positive Erdős number $n$ at time $t$. As old mathematicians leave, and new mathematicians come, how does $E(n,t)$ change over time ?
We can …
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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 …
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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 …
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