1-Factorization of K 2n

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1-Factorization of $K_{2n}$ --- один-факторизация графа $K_{2n}$.

A one-factorization of $K_{2n}$ is a partition of the edge-set of $K_{2n}$ into $2n-1$ one-factors. A perfect one-factorization (P1F) is a one-factorization in which every pair of distinct one-factors forms a Hamiltonian cycle of $K_{2n}$. P1Fs of $K_{2n}$ are known to exist when $2n-1$ or $n$ is prime, and for $2n \in \{16, 28, 36,40, 50, 126, 170, 244, 344, 730, 1332, 1370, 1850, 2198, 3126, 6860, 12168, 16808, 29792\}$.

It has been conjectured that a perfect one-factorization of $K_{2n}$ exists for all $n \geq 2$.