  It is conjectured that every edge-colored complete graph $G$ on $n$ vertices
satisfying $\Delta^{mon}(G)\leq n-3k+1$ contains $k$ vertex-disjoint properly
edge-colored cycles. We confirm this conjecture for $k=2$, prove several
additional weaker results for general $k$, and we establish structural
properties of possible minimum counterexamples to the conjecture. We also
reveal a close relationship between properly edge-colored cycles in
edge-colored complete graphs and directed cycles in multi-partite tournaments.
Using this relationship and our results on edge-colored complete graphs, we
obtain several partial solutions to a conjecture on disjoint cycles in directed
graphs due to Bermond and Thomassen.
