A conjecture on complete graphs

A defining set of 1-factorizations of a graph G is a set of partial 1-factors of G which may be completed to a unique 1-factorization of G. We construct defining sets of size n²-3n+4 in a 1-factorization of K_2n for each n ≤ 3 and give an (improved) upper bound for the minimum size of a defining set among all 1-factorizations of K_2n.