Abstract
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}).
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