# 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