All Questions
Tagged with graph-theory polyhedra
11 questions
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Growth polynomial of the Associahedron graph ? (Is it approximately Gaussian ?)
Consider Associahedron, consider graph build from its vertices and edges. Choose some vertex. Let us count the number of vertices on distances $k$ from the selected vertex. Write a generating ...
- 24.9k
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A face and all its neighbors: terminology?
Suppose $F$ is a face of a 2-complex, and $F_1,\dotsc,F_n$ are the faces that are adjacent to (i.e., share an edge with) $F$. Is there a standard term for a collection of faces of the form $\{F,F_1,\...
- 19.7k
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Distinguishable knots (with constraints) over polyhedra
I'm trying to find the number of distinguishable knot projections over certain convex regular polyhedra according to the following constraints. On each face on the polyhedron the knot will have a ...
- 2,558
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Clustering of vertices in an $n$-dimensional cube
Consider the vertices of an $n$-dimensional cube. The distance between two vertices is measured as the minimum number of edges between the two vertices. Now consider a subset of these vertices.
If we ...
- 41
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Reconstructing plane graphs from degree- and face-sequences
Let $G$ be a plane $3$-connected graph; so it partitions the plane
into regions bounded by faces.
Let $\mathrm{deg}_v$ be the sequence of vertex degrees of $G$,
and $\mathrm{deg}_f$ be the sequence of ...
- 151k
4
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Harborth's conjecture and polyhedra
Harborth's conjecture states that every planar graph can be drawn on a plane only using staight line segments of rational or integral edge length.
( There is a good mathoverflow page for this ...
- 481
4
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Convex polyhedra, combinatorial types and Symmetry
Steinitz theorem says that combinatorial types of convex polyhedra is identified with 3-connected planar graph, called by a polyhedral graph.
A symmetry of polyhedral graph means that a vertex ...
- 481
9
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3
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Labeling edges of an icosahedron with sum constraints
The question is inspired by this previous MO question. There it was shown that it's possible to label the edges of a cube by the numbers $\{1,2,\ldots,6,8,9, \ldots, 13\}$ in such a way that:
Three ...
- 20.8k
5
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regular polyhedra (and polytopes) in hyperbolic geometry, and generalisations
While there exist regular tesselations of the hyperbolic plane with arbitrary regular polygons, there are no new regular polyhedra in hyperbolic (3D) space. This being quite trivial, it is probably ...
- 3,630
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Visualizing polyhedra from their 1-skeletons
Except for a few simple cases (typically pyramids and prisms) I find it hard to visualize a polyhedron from its 1-skeleton embedded in the plane, e.g. the hexahedral graph 5, as can be seen here.
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- 9,720
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Does Euler's formula imply bounds on the degree of vertices in a 3-polytopal graph?
A corollary of Euler's formula tells us that the edge-vertex graph of every convex 3-polyhedron must have a face with either 3, 4, or 5 edges, using an argument about the average degree of vertices.
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