On a board like this, there are three simple ways to form a non-trivial loop, each of minimum length n:
. . x . . . . . . . . . . x . . . . . . x . . . . . . . . . x . . . . . . . x . . . . . . . . . . . . . . x . . x . . . . . . . . . . . . . x . . . x . . . x x x x x x . . . x . . . . x . . . . . . . . . . . x . . .
Vert Horz Diag
This three-player game has also recently been discovered by Bill Taylor (kiwibill) who calls it Porus Torus.
P. S. Your homework assignment (2 stars): Invent a connection game on the Klein bottle.
Divide the klein bottle surface into a finite number of similar cells (hexagons plus six pentagons, as per the sphere?). Moving in a cell colours the cell faces on both sides of the surface. Then either:
a) Play Projex; or b) Assign two maximally distant cells to one player and play Antipod.
These are the only two connection games I know of that are totally independent of edges.