Finite topologies and Hamiltonian paths

Suppose x is a finite set. This paper deals with the question of how many mutually complementary topologies X can carry. If p is a prime and | X |= p , p +1, 2p −1 or 2p , we prove that the answers are respectively p, p, 2p −1, 2p −1. The problem is shown to be related to the existence of a certain type of 1-factorization of the complete graph on an even number of points, and is also formulated combinatorially.