Minimal colorings for properly colored subgraphs in complete graphs

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 show that $pr(K_n,G)-ex(n,{\mathcal{G}}^{\mathcal{\prime }})=o(n^2),$ where $${\mathcal{G}}^{\mathcal{\prime }}$={G-M: M $\text{ is a matching of }$G}$\mathrm{.}Furthermore,wedeterminethevalueof$pr(K_{n}, P_{l})$for$l\ge 27$and$n\ge 2l^{3}$andtheexactvalueof$pr(K_{n}, G)$,where$G$is$C_{5}, C_{6}$and$K_{4}^{-}$, respectively. Also, we give an upper bound and a lower bound of $pr(K_n,K_{2,3})$.