1-Factorization of K 2n: различия между версиями
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distinct one-factors forms a Hamiltonian cycle of <math>K_{2n}</math>. P1Fs of | distinct one-factors forms a Hamiltonian cycle of <math>K_{2n}</math>. P1Fs of | ||
<math>K_{2n}</math> are known to exist when <math>2n-1</math> or <math>n</math> is prime, and for <math>2n | <math>K_{2n}</math> are known to exist when <math>2n-1</math> or <math>n</math> is prime, and for <math>2n | ||
\in \{16, 28, 36,40, 50, 126, 170, 244, 344, 730, | \in \{16, 28, 36,40, 50, 126, 170, 244, 344, 730, 1332, 1370, 1850, | ||
2198, 3126, 6860, 12168, 16808, 29792\}</math>. | 2198, 3126, 6860, 12168, 16808, 29792\}</math>. | ||
It has been conjectured that | It has been conjectured that | ||
a perfect one-factorization of <math>K_{2n}</math> exists for all <math>n \geq 2</math>. | a perfect one-factorization of <math>K_{2n}</math> exists for all <math>n \geq 2</math>. | ||
Текущая версия от 13:58, 24 сентября 2018
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].