New families of atomic Latin squares and perfect 1-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 Kp+1 (one of which is well known) and five non-isomorphic perfect 1-factorisations of Kp,p. If 2 is a primitive root modulo p, then we show the existence of 11 non-isomorphic perfect 1-factorisations of Kp,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.