  We prove two results regarding cycles in properly edge-colored graphs.
  First, we make a small improvement to the recent breakthrough work of Alon,
Pokrovskiy and Sudakov who showed that every properly edge-colored complete
graph $G$ on $n$ vertices has a rainbow cycle on at least $n - O(n^{3/4})$
vertices, by showing that $G$ has a rainbow cycle on at least $n - O(\log n
\sqrt{n})$ vertices.
  Second, by modifying the argument of Hatami and Shor which gives a lower
bound for the length of a partial transversal in a Latin Square, we prove that
every properly colored complete graph has a Hamilton cycle in which at least $n
- O((\log n)^2)$ different colors appear.
  For large $n$, this is an improvement of the previous best known lower bound
of $n - \sqrt{2n}$ of Andersen.
