## Abstract

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 K_{6}/3 and K_{7}/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.

Original language | English |
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Pages (from-to) | 243-260 |

Number of pages | 18 |

Journal | Transactions of the American Mathematical Society |

Volume | 242 |

DOIs | |

Publication status | Published - 1978 |

Externally published | Yes |

## Keywords

- Complete graphs
- Factorisations