--so if I understand aright, you took the 6x6 torus chessboard, and identified antipodal squares, thus getting 18 = (6*6/2) vertices, then used knight moves to define edges? Thus getting a graph of valency 4? Wrong. This would yield valency 8.

No, it wouldn't. (+1, +2) = (-2, -1) modulo (3, 3), so pairs of knight moves would land on the same square (after `projectivisation' (quotienting out by identifying `antipodal' squares)). Have I really outwitted Warren whilst under the influence of three glasses of Cotes-du-Rhone? :) Sincerely, Adam P. Goucher