One-factorisations of complete graphs constructed in desarguesian planes of certain odd square orders

A geometric construction of one-factorisations of complete graphs Kq(q−1) is provided for the case when either q = 2d + 1 is a Fermat prime, or q = 9. This construction uses the affine group AGL(1, q), points and ovals in the Desarguesian plane PG(2, q2) to produce one-factorisations of the complete graph Kq(q−1)