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

It is conjectured that every edge-colored complete graph (Formula presented.) on (Formula presented.) vertices satisfying (Formula presented.) contains (Formula presented.) vertex-disjoint properly edge-colored cycles. We confirm this conjecture for (Formula presented.), prove several additional weaker results for general (Formula presented.), 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 multipartite 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