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

Source abstract

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