Graphs that
Quadrangulate both the Torus and Klein Bottle
A quadrangulation of a
surface S is an embedded
graph G such that every face
is of size 4. A diamond 2-curve
is a curve C in S such that C intersects the quadrangulation in
a pair of vertices {x,y}.
Hence all but two points on the curve lie in the interior of two faces xayb, ycxd. If C is non-contractible, then it is
called an essential diamond
2-curve.
If G quadrangulates S and has an essential diamond
2-curve, then it is easy to cut along this curve and re-imbed G so that it quadrangulates the
Klein bottle (this embedding has faces xayd and xcyb). Nakamoto conjectures that
this sufficient condition is necessary:
Conjecture: If G is a graph that has a
quadrangular embedding in the torus, then G has a quadrangular
embedding in the Klein bottle if and only if the toroidal embedding has
an essential diamond 2-curve.
References:
[N] A. Nakamoto, Four-regular graphs which quadrangulate both the torus and the Klein bottle, (preprint).
Submitted by: Dan Archdeacon (with thanks to Atsuhiro Nakamoto nakamoto@edhs.ynu.ac.jp)
Send comments to dan.archdeacon@uvm.edu
November, 2003