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.

中文翻译：

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边着色完全图中顶点不相交的正确边着色循环

推测满足$\Delta^{mon}(G)\leq n-3k+1$的$n$个顶点上的每个边着色完全图$G$包含$k$个顶点不相交的正确边着色循环。我们证实了 $k=2$ 的这个猜想，证明了一般 $k$ 的几个额外的较弱的结果，并且我们建立了该猜想的可能最小反例的结构特性。我们还揭示了边着色完整图中的适当边着色循环与多方锦标赛中的有向循环之间的密切关系。使用这种关系和我们在边缘着色完全图上的结果，我们获得了对由于 Bermond 和 Thomassen 导致的有向图中不相交环的猜想的几个部分解。