1
vote
0answers
34 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 ...
1
vote
1answer
27 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 ...
5
votes
0answers
67 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 ...
2
votes
1answer
168 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. ...
1
vote
1answer
86 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 ...
5
votes
1answer
117 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 ...
1
vote
1answer
46 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 ...
2
votes
1answer
66 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 $\mathbb{R}^{3v}$, ...
1
vote
1answer
183 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 ...
1
vote
1answer
89 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: ...
2
votes
3answers
97 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 ...
3
votes
3answers
121 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 ...