New families of atomic Latin squares and perfect one-factorisations

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 eleven non-isomorphic perfect 1-factorisations of K{p,p} and five 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.
Last modified: Tue Sep 7 19:06:43 EST 2004