Almost all optimally coloured complete graphs contain a rainbow Hamilton path

TL;DR

This paper proves that almost all optimal edge-colourings of complete graphs contain a rainbow Hamilton path and cycle, confirming Andersen's conjecture for most cases and linking to Latin square transversals.

Contribution

It demonstrates that nearly all optimal colourings of complete graphs contain rainbow Hamilton paths and cycles, confirming Andersen's conjecture in a broad setting.

Findings

Almost all optimal colourings of $K_{n}$ have rainbow Hamilton paths.

Almost all optimal colourings of $K_{n}$ contain rainbow cycles using all colours.

The results have implications for transversals in random Latin squares.

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.