Vertex-disjoint properly edge-colored cycles in edge-colored complete graphs
Abstract
It is conjectured that every edge-colored complete graph G on n vertices satisfying mon(G)≤ 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.
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