On the Maximal Colorings of Complete Graphs Without Some Small Properly Colored Subgraphs (original) (raw)

Let \mathrm{pr}(K_{n}, G)pr(Kn,G)bethemaximumnumberofcolorsinanedge−coloringofpr ( K n , G ) be the maximum number of colors in an edge-coloring ofpr(Kn,G)bethemaximumnumberofcolorsinanedge−coloringofK_{n}KnwithnoproperlycoloredcopyofG.ForafamilyK n with no properly colored copy of G. For a familyKnwithnoproperlycoloredcopyofG.Forafamily{\mathcal {F}}Fofgraphs,letF of graphs, letFofgraphs,let\mathrm{ex}(n, {\mathcal {F}})ex(n,F)bethemaximumnumberofedgesinagraphGonnverticeswhichdoesnotcontainanygraphsinex ( n , F ) be the maximum number of edges in a graph G on n vertices which does not contain any graphs inex(n,F)bethemaximumnumberofedgesinagraphGonnverticeswhichdoesnotcontainanygraphsin{\mathcal {F}}Fassubgraphs.Inthispaper,weshowthatF as subgraphs. In this paper, we show thatFassubgraphs.Inthispaper,weshowthat\mathrm{pr}(K_{n}, G)-\mathrm{ex}(n, \mathcal {G'})=o(n^{2}), pr(Kn,G)−ex(n,G′)=o(n2),wherepr ( K n , G ) - ex ( n , G ′ ) = o ( n 2 ) , wherepr(Kn,G)−ex(n,G′)=o(n2),where\mathcal {G'}=\{G-M: M \text { is a matching of }G\}G′=G−M:MisamatchingofG.Furthermore,wedeterminethevalueofG ′ = { G - M : M is a matching of G } . Furthermore, we determine the value ofG′=G−M:MisamatchingofG.Furthermore,wedeterminethevalueof\mathrm{pr}(K_{n}, P_{l})pr(Kn,Pl)forsufficientlylargenandtheexactvalueofpr ( K n , P l ) for sufficiently large n and the exact value ofpr(Kn,Pl)forsufficientlylargenandtheexactvalueof\mathrm{pr}(K_{n}, G)pr(Kn,G),whereGispr ( K n , G ) , where G ispr(Kn,G),whereGisC_{5}, C_{6}C5,C6andC 5 , C 6 andC5,C6andK_{4}^{-}K4−,respectively.Also,wegiveanupperboundandalowerboundofK 4 - , respectively. Also, we give an upper bound and a lower bound ofK4−,respectively.Also,wegiveanupperboundandalowerboundof\mathrm{pr}(K_{n}, K_{2,3})$$ pr ( K n , K 2 , 3 ) .