The perfect 1-factorisation conjecture holds asymptotically

Abstract

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.