Black 1-factors and Dudeney sets

A set of Hamilton cycles in the complete graph K_n is called a Dudeney set if every path of length two lies on exactly one of the cycles. It has been conjectured that there is a Dudeney set for every complete graph. It is known that there exists a Dudeney set for K_n when n is even, but the question is still unsettled when n is odd. In this paper, we define a black 1-factor in K_p+1 for an odd prime p, and show that if there exists a black 1-factor in K_p+1, then we can construct a Dudeney set for K_p+2. We also show that if there is a black 1-factor in K _p+1, then 2 is a quadratic residue modulo p. Using this result, we obtain some new Dudeney sets for K_n when n is odd.