Re: Labelling polyhedron faces
- From: "Proginoskes" <CCHeckman@xxxxxxxxx>
- Date: 18 Oct 2006 15:28:26 -0700
Axel Harvey wrote:
For example, how many ways are there to label the
faces of a regular dodecahedron with numbers n
from 0 to 11 so that (1) all pairs of faces n and n+1
share an edge; (2) face n is diametrically opposed
to face n+6 mod 12; (3) faces 11 and 0 share an
edge?
I'm not so much interested in a specific answer (I
think I can get it with a model and post-it stickers)
as in approaches to this sort of problem. I am not a
mathematician so I don't even know what general
topic it falls under.
Graph Theory. You're looking for a Hamiltonian cycle in the icosohedron
graph.
To get the other condition (n is opposite n+6), you actually want to
look at the graph H obtained from the icosohedron by identifying
opposite pairs of vertices. Then any Hamiltonian cycle of H will give
you a Hamiltonian cycle of the icosohedron where n is opposite n+6.
H might be a little difficult to describe, so I might follow up on
this. (I have to rush off and teach class.)
--- Christopher Heckman
.
- References:
- Labelling polyhedron faces
- From: Axel Harvey
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