A subgraph H of an edge-coloured graph is called rainbow if all of the edges of H have different colours. In 1989, Andersen conjectured that every proper edge-colouring of $K_n$ admits a rainbow path of length $n-2$. We show that almost all optimal edge-colourings of $K_n$ admit both (i) a rainbow Hamilton path and (ii) a rainbow cycle using all of the colours. This result demonstrates that Andersen's Conjecture holds for almost all optimal edge-colourings of $K_n$ and answers a recent question of Ferber, Jain, and Sudakov. Our result also has applications to the existence of transversals in random symmetric Latin squares.

中文翻译：

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几乎所有最优着色的完全图都包含一条彩虹汉密尔顿路径

如果H的所有边都具有不同的颜色，则边色图的子图H称为彩虹。1989 年，安徒生猜想每一个适当的边缘着色${ķ}_n$承认彩虹的长度$n-2$. 我们证明了几乎所有的最优边缘着色${ķ}_n$承认（i）彩虹汉密尔顿路径和（ii）使用所有颜色的彩虹循环。这一结果表明，安徒生猜想几乎适用于所有最优边缘着色${ķ}_n$并回答了 Ferber、Jain 和 Sudakov 最近提出的问题。我们的结果也适用于随机对称拉丁方格中横向的存在。