On the Maximal Colorings of Complete Graphs Without Some Small Properly Colored Subgraphs

Let $$\mathrm{pr}(K_{n}, G)$$ be the maximum number of colors in an edge-coloring of $$K_{n}$$ with no properly colored copy of G. For a family $${\mathcal {F}}$$ of graphs, let $$\mathrm{ex}(n, {\mathcal {F}})$$ be the maximum number of edges in a graph G on n vertices which does not contain any graphs in $${\mathcal {F}}$$ as subgraphs. In this paper, we show that $$\mathrm{pr}(K_{n}, G)-\mathrm{ex}(n, \mathcal {G'})=o(n^{2}), $$ where $$\mathcal {G'}=\{G-M: M \text { is a matching of }G\}$$ . Furthermore, we determine the value of $$\mathrm{pr}(K_{n}, P_{l})$$ for sufficiently large n and the exact value of $$\mathrm{pr}(K_{n}, G)$$ , where G is $$C_{5}, C_{6}$$ and $$K_{4}^{-}$$ , respectively. Also, we give an upper bound and a lower bound of $$\mathrm{pr}(K_{n}, K_{2,3})$$ .