All Questions
Tagged with planar-graphs polyhedra
13
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How high the connectivity of a planar graph can ensure that any two faces share at most two vertices?
Once I asked the following question.
Question 1 (solved): Do any two faces share with at most one edge in a 3-connected plane graph?
Do two faces of any 3-connected planar graph have at most one ...
- 2,474
1
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1
answer
84
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Finding the Canonical Polyhedron associated with a 3-connected simple graphs.
I am not a professional mathematician but I am a reasonably competent programmer and I am also no stranger mathematics, though I must say that my usual domain is closer to calculus and functions ...
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Is there any new developments on the Barnette's conjecture?
When I searching for interesting math problems. I find there is a graph theory conjecture called the Barnette's conjecture.
The statement is: Is every bipartite simple polyhedron Hamiltonian?
A early ...
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1
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2
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Maximum number of edges in a concave polyhedron given n vertices
Given $n$ vertices of a concave polyhedron (3D), what are the maximum amount of edges it can have?
I know for convex polyhedra the upper bound is $3n-6$. Does this also hold for concave polyhedra?
...
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1
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Existence of planar graph whose faces correspond to the faces of a convex polyhedron
Wikipedia states that Steinitz's theorem says:
"a given graph $G$ is the graph of a convex three-dimensional
polyhedron, if and only if $G$ is planar and $3$-vertex-connected"
So, given a convex ...
- 3,415
2
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3
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165
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Are there highly symmetric polyhedra in which most of the vertices are of degree seven?
Are there highly symmetric polyhedra in which most of the vertices are of degree seven?
I realize that this question is vague, so I'll provide some more context. I have recently been doing research ...
user253970
3
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2
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Converse of Euler's formula for polyhedra
Is it true that for any positive integers $V, E, F$ with $V - E + F = 2$ there exists a polyhedron with $V$ vertices, $E$ edges and $F$ faces?
In case there is a silly counterexample (say, with $F=1$)...
- 2,793
2
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1
answer
339
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A convex polyhedron has 20 vertices and 12 faces. Each face of the polyhedron is bounded by the same number of edges. What is this common number?
A convex polyhedron has 20 vertices and 12 faces. Each face of the polyhedron is bounded by the
same number of edges. What is this common number?
If I am not mistaken , "this common number" is the ...
- 107
7
votes
3
answers
512
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Are there only seven 3-connected planar graphs with certain symmetries?
There are only five regular polyhedra, and only two more if we allow for quasi-regular polyhedra. So in the sum, there are seven (convex) polyhedra which are both vertex- and edge-transitive.
Since ...
- 30.1k
2
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1
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The skeleton of Eulerian polyhedra
There is (at least) two kind of validity domain of Euler's $v−e+f=2$ polyhedron formula. One is the "Eulerian" polyhedra, i.e. simply connected polyhedra with simply connected faces (see here). The ...
- 2,065
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Proving the upper bound of edges in a convex polyhedron
The question is the following:
Suppose Every face of a convex polyhedron has at least $5$ vertices and every vertex has degree $3$. Prove that if the number of vertices is $n$, then the number of ...
- 359
2
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Does every polyhedral graph have a path cover with non-empty paths?
I'm looking to prove or disprove the following conjecture:
Every polyhedral graph has a path cover with vertex disjoint, non-zero (length $\ge 1$) paths.
Any pointers to literature are appreciated.
...
11
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1
answer
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Euler's formula for triangle mesh
Can anyone explain to me these two facts which I don't get from Euler's formula for triangle meshes?
First, Euler's formula reads $V - E + F = 2(1-g)$ where $V$ is vertices number, $E$ edges number, $...
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