Sextic Toroidal Graphs

On a plane, four nodes can be connected by edges without any edges crossing.
On a torus, seven nodes can be connected by edges with each node connecting to the other six, again without any edges crossing. A toroidal mapping can be represented by flattening the torus and the graph onto a square such that the left and right sides are identified and the top and bottom sides are identified.
Of the first 39 sextic toroidal graphs, all but two are circulant graphs on vertices where each vertex connects to vertices . Of those 37 graphs, 33 of them have the property that . For these 33 graphs, if each edge is labeled with the modular difference of the vertex labels, then the edge-labeled triangles are all similar. This allows the placement of points on an grid at positions . The other six graphs can also be represented as points and lines. This Demonstration shows flattened toroidal mappings of these graphs.

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