**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:

Submitted by: Dan Archdeacon (with thanks to Atsuhiro Nakamoto *nakamoto@edhs.ynu.ac.jp*)

Send comments to *dan.archdeacon@uvm.edu*

November, 2003