Isomorphic factorisations. I: Complete graphs

An isomorphic factorisation of the complete graph K_p is a partition of the lines of K_p into t isomorphic spanning subgraphs G; we then write GK_p and G e K_p/t. If the set of graphs K_p/t is not empty, then of course t\p(p - 1)/2. Our principal purpose is to prove the converse. It was found by Laura Guidotti that the converse does hold whenever (t, p) = 1 or (t, p - 1) = 1 We give a new and shorter proof of her result which involves permuting the points and lines of K_p. The construction developed in our proof happens to give all the graphs in K₆/3 and K₇/3. The Divisibility Theorem asserts that there is a factorisation of K_p into t isomorphic parts whenever t divides p(p - 1)/2. The proof to be given is based on our proof of Guidotti's Theorem, with embellishments to handle the additional difficulties presented by the cases when t is not relatively prime to p or p - 1.