1-Factorization of K 2n

1-Factorization of [math]\displaystyle{ K_{2n} }[/math] --- один-факторизация графа [math]\displaystyle{ K_{2n} }[/math].

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

It has been conjectured that a perfect one-factorization of [math]\displaystyle{ K_{2n} }[/math] exists for all [math]\displaystyle{ n \geq 2 }[/math].