## Abstract

A perfect 1 -factorisation of a graph G is a decomposition of G into edge disjoint 1 -factors such that the union of any two of the factors is a Hamiltonian cycle. Let p ≥ 11 be prime. We demonstrate the existence of two non-isomorphic perfect 1-factorisations of K_{p+1} (one of which is well known) and five non-isomorphic perfect 1-factorisations of K_{p,p}. If 2 is a primitive root modulo p, then we show the existence of 11 non-isomorphic perfect 1-factorisations of K_{p,p} and 5 main classes of atomic Latin squares of order p. Only three of these main classes were previously known. One of the two new main classes has a trivial autotopy group.

Original language | English |
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Pages (from-to) | 608-624 |

Number of pages | 17 |

Journal | Journal of Combinatorial Theory. Series A |

Volume | 113 |

Issue number | 4 |

DOIs | |

Publication status | Published - May 2006 |