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arXiv · 2007.00395

Almost all optimally coloured complete graphs contain a rainbow Hamilton path

Stephen Gould·Tom Kelly·Daniela Kühn·Deryk Osthus
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Abstract

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.

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Stephen Gould, Tom Kelly, Daniela Kühn, Deryk Osthus. 2020-07-01. Almost all optimally coloured complete graphs contain a rainbow Hamilton path. https://arxiv.org/abs/2007.00395

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