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\geq 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 eleven non-isomorphic
perfect 1-factorisations of Kp,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