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polyhedra/graph question
Posted:
Jan 27, 1999 2:03 PM
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Ages ago, some one asked on this list (I think)
for a description of all possible planar graphs with all
vertices degree 3, and all faces having 5 or 6
sides (so you can start with a dodecahedron).
Did anyone get round to finding the solution yet?
I was just wondering about this again recently;
there are about 4 pairs (n,a,b) such that there are
infinitely many graphs with n the degree of each
vertex, and a, b the possible numbers of sides of a face.
for each possible set (n,a,b), I can find loads of infinite
families, but I've no idea how to get everything.
Here's a nice little thing: if you have degree 4
vertices, and faces are either squares or triangles,
show that you have to have an even number of vertices.
I'd like to know - is this a topological property,
or something else? (Cos I didn't prove with with
Euler, but it would be nice to know if there was a
'topological' way to prove it, and if it has generalizations, etc)
Helena
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