Abstract
Let G be an edge-colored complete graph on n vertices such that there exist at least n distinct colors on edges incident to every pair of its vertices. In this paper, we first show that every edge of G with n≥6k−19 is contained in a properly colored cycle of length k. Further, we prove that if G contains no monochromatic triangles, then there exists a properly colored path of length l for every 1≤l≤n−1 between each pair of vertices of G and every vertex of G is contained in a properly colored cycle of length k for any 3≤k≤n.
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