Long directed rainbow cycles and rainbow spanning trees

A subgraph of an edge-coloured graph is called rainbow if all its edges have different colours. The problem of finding rainbow subgraphs goes back to the work of Euler on transversals in Latin squares and was extensively studied since then. In this paper we consider two related questions concerning rainbow subgraphs of complete, edge-coloured graphs and digraphs. In the first part, we show that every properly edge-coloured complete directed graph contains a directed rainbow cycle of length $n-O(n^{4/5})$. This is motivated by an old problem of Hahn and improves a result of Gyarfas and Sarkozy. In the second part, we show that any tree $T$ on $n$ vertices with maximum degree $\Delta_T\leq \beta n/\log n$ has a rainbow embedding into a properly edge-coloured $K_n$ provided that every colour appears at most $\alpha n$ times and $\alpha, \beta$ are sufficiently small constants.