Minimal colorings for properly colored subgraphs in complete graphs

Abstract

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'})=o(n^{2}), $ where $\mathcal{G'}=\{G-M: M \text{ is a matching of }G\}$. Furthermore, we determine the value of $pr(K_{n}, P_{l})$ for $l\ge 27$ and $n\ge 2l^{3}$ and the exact value of $pr(K_{n}, G)$, where $G$ is $C_{5}, C_{6}$ and $K_{4}^{-}$, respectively. Also, we give an upper bound and a lower bound of $pr(K_{n}, K_{2,3})$.