Properly colored cycles in edge-colored complete graphs

As an analogy of the well-known anti-Ramsey problem, we study the existence of properly colored cycles of given length in an edge-colored complete graph. Let pr(K-n,G) be the maximum number of colors in an edge-coloring of K-n with no properly colored copy of G. In this paper, we determine the exact threshold for cycles pr(K-n,C-l), which proves a conjecture proposed by Fang, Gyori, and Xiao, that the maximum number of colors in an edge-coloring of K-n with no properly colored copy of C-l is max{((l-1)(2))+n-l+1,left perpendicular(l-1)/(3)right perpendicularn-(left perpendicularl(-1)/(3)right perpendicular(2)(+1))+1+r(l-1)}, where Cl is a cycle on l vertices, l-1 equivalent to r(l-1) mod 3, and 0 <=(l-1)<= 2. It is a slight modification of a previous conjecture posed by Manoussakis, Spyratos, Tuza and Voigt. Also, we consider the maximal coloring of K-n whether a properly colored cycle can be extended by exact one more vertex.