Vertex-disjoint properly edge-colored cycles in edge-colored complete graphs

It is conjectured that every edge-colored complete graph $G$ on $n$ vertices satisfying ${\mathrm{\Delta }}^{mon}(G)\le n-3k+1$ 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.