  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.
