# Tagged Questions

1answer
83 views

### Which polyhedron has 17 vertices, 34 edges and 19 faces?

on exam I had task to check that there is polyhedron with 8 triangle faces, 11 quadrangle, each vertices have degree 4. after calculate I obtain that it have 34 edges, 17 vertices and 19 faces but i ...
0answers
38 views

### showing that all convex polehedron graphs are 3-connected

I'm trying to figure out how to show that two nonadjacent vertices in the graph of a convex polyhedron can be disconnected from one another by the removal of at least three vertices. I know what a ...
1answer
28 views

### Give examples of polytopes $\Delta$ in $\mathbb{AR}^3$ such that

With Sym $\Delta$ of the set $\Delta$ consisting of all isometries of $\mathbb{AR}^n$ that map $\Delta$ onto $\Delta$, Sym $\Delta$ acts transitively on the set of vertices of $\Delta$ but is ...
0answers
77 views

### Is Paley-13 graph a unit distance graph in 3D space?

The 13-node Paley graph has vertices 1 to 13 that are connected by an edge when their difference is one of the values $(1,3,4,9,10,12)$ Can this graph be put into 3D space so that all edges have ...
1answer
202 views

### Number of edge colorings in a tetrahedron with three colors. Is my solution correct?

I've tried to count rotationally distinct edge colorings (both proper and improper) in a regular tetrahedron with three colors. Could you take a look if it's correct? First the improper colorings. ...
1answer
89 views

### Is this graph coloring problem solved correctly?

On this Wikipedia page about Burnside's lemma, it is calculated that there are 57 rotationally distinct colorings of the faces of a cube with three colors. I'm confused by the way it is done. They ...
1answer
119 views

### Is There a Formalization of Cauchy's $F - E+V = 2$ proof?

Can anyone provide, or direct me to a formalized version of Cauchy's proof that for any convex polyhedron with $F$ faces, $E$ edges and $V$ vertices that $F - E + V = 2$. I am willing to accept the ...
1answer
58 views

### Part from “Regular polytopes” which I don't understand

This is a paragraph from "Regular polytopes" by Coxeter that I don't understand. Although it is not always possible to include all the vertices of a polyhedron in a single chain of edges, it ...
1answer
69 views

### The structure of realization spaces of polyhedral graphs

Given a polyhedral graph with $v$ vertices, $e$ edges and $f$ faces, each possible realization of the graph as a geometric (convex) polyhedron corresponds to a point in $\mathbb{R}^{3v}$, ...
1answer
199 views

### Graph Isomorphisms, Delaunay Triangulation on a sphere, and Kulikowski's Theorem

Suppose I have a collection of $n$ non-collinear points on a sphere, $\left\lbrace P_i\right\rbrace_{i=1}^n$. And I construct a mapping from this collection of points to the Delaunay Triangulation ...
2answers
103 views

### Polyhedra from Cayley Graphs

I was playing around with the Cayley graphs for some simple groups today and stumbled across something interesting, but can't quite figure out if there's something deeper going on. Here's what I did: ...
3answers
100 views

### Finding the number of edges that connect to a single vertex in a dodecahedron

Please note my geometry background is very weak (high school geometry is all I have), so I would appreciate it if someone could explain it in very layman terms how to do this. I am trying to solve ...
3answers
131 views

### Intuition about the faces in the connected planar graphs

In the Euler formula, for counting the number of faces, we count the regions bounded by edges, including the outer, infinitely-large region, so in the graph $K_1$ there is only one face which is outer ...