AI 中文总结
研究Anton Kotzig关于完全图Kn的完美1-因子分解猜想，通过证明Kn可分解为n - 1个完美匹配，其中(1 - o(1))n个匹配中任意一对都能形成哈密顿圈，得出该猜想渐近成立的结论。
AI 中文摘要
Anton Kotzig的一个著名猜想指出，对于每个大于等于4的偶数整数n，阶为n的完全图Kn可以分解为n - 1个完美匹配，使得这些匹配中的每一对都形成一个哈密顿圈。尽管备受关注，但该猜想远未解决。本文表明该猜想渐近成立，即Kn可分解为n - 1个完美匹配，其中(1 - o(1))n个匹配具有任意一对都形成哈密顿圈的性质。
英文摘要
A famous conjecture of Anton Kotzig states that for every even integer $n\ge 4$, the complete graph $K_n$ of order $n$ can be decomposed into $n - 1$ perfect matchings such that every pair of these matchings forms a Hamilton cycle. Despite the great interest, the conjecture is far from being solved. Here we show that the conjecture holds asymptotically, namely that $K_n$ can be decomposed into $n-1$ perfect matchings such that $(1-o(1))n$ of them have the property that any pair forms a Hamilton cycle.
Comments6 pages, 2 figures
