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Results tagged with graph-theory
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user 6094
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.
22
votes
Interesting and accessible topics in graph theory
I have found that the Art Gallery Problem engages middle- and high-school students, and quickly leads to the unknown, which itself can be eye-opening to students. (On the latter point, students tend t …
15
votes
Accepted
Travelling Salesman Problem: Can the nearest neighbor algorithm be $n$ times longer than the...
The nearest-neighbor (NN) heuristic (among others) is analyzed in this paper:
Johnson, David S., and Lyle A. McGeoch. "The traveling salesman problem: A case study in local optimization." Local se …
- 151k
14
votes
Always a planar-drawn cycle through $n$ points
Here is a quote from the first paper cited below: Steinhaus posed a version of your question, which has become known as simple polygonization of a set of points:
1Agarwal, Pankaj K., Ferran …
- 151k
13
votes
Accepted
Difference Sets
A keyword in this area is homometric, and a key paper this one:
Joseph Rosenblatt and Paul D. Seymour.
"The Structure of Homometric Sets."
SIAM. J. on Algebraic and Discrete Methods, 3, 343-35 …
- 151k
12
votes
Accepted
Which degree sequences are planar graphical?
It is always difficult to say what is "currently known," but at least around
2008, the paper
"A Characterization of the degree sequences of 2-trees."
Prosenjit Bose, Vida Dujmovi, Danny Krizanc, …
- 151k
12
votes
Accepted
Shortest Path in Plane
The problem you posed is known in the literature as the weighted region problem.
It was the focus of Joe Mitchell's Ph.D. thesis, under the direction of
Papadimitriou, and their results were eventuall …
- 151k
12
votes
Embedding of planar graphs
The recent paper below (and its references) may help.
They mention that
every planar graph with max degree $4$ (except for the octahedron) admits
a $2$-bend embedding.
Deciding whether a graph can b …
- 151k
10
votes
10
votes
Concepts in topology successfully transferred to graph theory and combinatorics with non-tri...
There has been very interesting work on defining curvature on a discrete graph.
For example:
Knill, Oliver. "A graph theoretical Gauss-Bonnet-Chern theorem." arXiv:1111.5395 (2011). (Abstract.)
…
10
votes
A tree with prime vertices
Not an answer, just a drawing of the tree including the
OP's $2 \rightarrow 191$ path:
- 151k
10
votes
Which curves and surfaces are realizable by linkages? references?
Erik Demaine and I also included a proof for $d=2$ in Geometric Folding Algorithms:
Linkages, Origami, Polyhedra, Chapter 3.
There we asked if there is a planar (non-crossing) linkage that
"signs you …
- 151k
9
votes
Accepted
Elegant representations of graphs in R^3
You might start by exploring the various tools that are available.
For example, Mathematica's GraphPlot3D[] does a nice job with small graphs.
Here is a 5-node graph:
And here is a somewhat den …
- 151k
9
votes
Visualizing polyhedra from their 1-skeletons
In response to the request for "a visualization of the hexahedral graph 5":
Just to illustrate the point that there are multiple realizations of any polyhedral graph:
- 151k
9
votes
Accepted
Smallest Connected Graph for Given Degree Sequence
A theorem of Hakimi says
that any pair of degree-equivalent graphs can be obtained one
from the other by a sequence of "elementary $2$-switchings"
(probably known under many other names), which involv …
- 151k
8
votes
Generalizations of the four-color theorem
There is a recent generalization to $k$-uniform hypergraphs that are
embeddable in $\mathbb{R}^d$ without edge intersections.
"For $k=d=2$ the problem specializes to graph planarity":
Carl Georg Heis …