One-factorisations of complete graphs arising from ovals in finite planes

In this paper we use the geometry of finite planes to set up a procedure for the construction of one-factorisations of the complete graph. Let π be a projective plane of order n−1 with n even containing an oval Ω, and regard Ω as the vertex set of the complete graph Kn. Then any one-factorisation of Kn has a representation by a partition of the external points to Ω whose components are of size n and meet every tangent to 2 Ω in a unique point. Our goal is to construct such partitions from nice geometric configurations in the Desarguesian plane of order q with q=ph and p>2 prime.