Properly colored cycles of different lengths in edge-colored complete graphs

A cycle in an edge-colored graph is called properly colored if all of its adjacent edges have distinct colors. Let K_n^c be an edge-colored complete graph with n vertices and let k be a positive integer. Denote by Δ^mon(K_n^c) the maximum number of edges of the same color incident with a vertex of K_n^c. In this paper, we show that (i) if Δ^mon(K_n^c)⩽n−2k, then K_n^c contains k properly colored cycles of different lengths and the bound is sharp; (ii) if Δ^mon(K_n^c)⩽n−2^k+1−2k+4, then K_n^c contains k vertex-disjoint properly colored cycles of different lengths; in particular, Δ^mon(K_n^c)⩽n−6 suffices for the existence of two vertex-disjoint properly colored cycles of different lengths.