Simple perfect square-cylinders of low order

We report the existence of two simple perfect square-cylinders of order 20 shown in Figs. 1 and 2. Dissected cylinders are associated with graphs which we call cyl-nets. A cyl-net is obtained from a dissected cylinder in the following way. The two rims of the cylinder are associated with two vertices which are called “source” and “sink,” respectively. Each horizontal line is associated with a vertex, while each square element is associated with an edge. Since a dissected cylinder can be embedded on a sphere the associated cylnet is planar. In case the cyl-net is 3-connected it must be a c-net. We are not interested in l-connected graphs since they are associated with compound dissections; that is, two cylinders on top of each other. The class of 2connected cyl-nets needs of course to be considered. This class can be constructed out of two cyl-nets of lower order. The order of a cylnet is the number of square elements of the cylinder. Let the first cyl-net have a source SO1 and a sink SIl, and let il be a vertex adjacent with SIl, but different from SO 1. The second cyl-net has a source S02, a sink S12 and a vertex i2 adjacent to S12, but different from S02. Then a new cyl-net can be obtained by identifying il with i2 and SIl with SI2 and removing either the edge (il, SIl) or edge (i2, S12). The new source is Sol, the new sink is S02. Another cyl-net can be obtained by removing both edges (il, Xl) and (i2, S12). In the same way one can identify il with SI2 and i2 with SIl, thus 333 0095.8956/83 $3.00