Isomorphic Factorisations . I : Complete Graphs

An isomorphic factorisation of the complete graph Kj, is a partition of the lines of Kp into t isomorphic spanning subgraphs G; we then write G\Kj, and G G K^/t. If the set of graphs Ky/t is not empty, then of course t\p(p — l)/2. Our principal purpose is to prove the converse. It was found by Laura Guidotti that the converse does hold whenever (/,/>) » 1 or (t,p — 1) «= 1, We give a new and shorter proof of her result which involves permuting the points and lines of Kp. The construction developed in our proof happens to give all the graphs in K6/3 and K-,/3. The Divisibility Theorem asserts that there is a factorisation of JÇ into t isomorphic parts whenever / dividesp(p — l)/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 top orp — 1.