Abstract
An isomorphic factorisation of the complete graphKp{K_p}is a partition of the lines ofKp{K_p}intotisomorphic spanning subgraphsG; we then writeG|KpG|{K_p}, andG∈Kp/tG \in {K_p}/t. If the set of graphsKp/t{K_p}/tis not empty, then of courset|p(p−1)/2t|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(t,p) = 1or(t,p−1)=1(t,p - 1) = 1. We give a new and shorter proof of her result which involves permuting the points and lines ofKp{K_p}. The construction developed in our proof happens to give all the graphs inK6/3{K_6}/3andK7/3{K_7}/3. The Divisibility Theorem asserts that there is a factorisation ofKp{K_p}intotisomorphic parts whenevertdividesp(p−1)/2p(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 whentis not relatively prime toporp−1p - 1.
Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
