A Family of Perfect Factorisations of Complete Bipartite Graphs

A 1-factorisation of a graph is perfect if the union of any two of its 1-factors is a Hamiltonian cycle. Let n=p2 for an odd prime p. We construct a family of (p−1)/2 non-isomorphic perfect 1-factorisations of Kn, n. Equivalently, we construct pan-Hamiltonian Latin squares of order n. A Latin square is pan-Hamiltonian if the permutation defined by any row relative to any other row is a single cycle.