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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.