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

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 . For a family ℱ of graphs, let ex(n, ℱ) be the maximum number of edges in a graph G on n vertices which does not contain any graphs in ℱ as subgraphs. In this paper, we show that pr(K_n, G)-ex(n, 𝒢')=o(n^2), where 𝒢'={G-M: M is a matching of G} . Furthermore, we determine the value of pr(K_n, P_l) for sufficiently large n 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) .