The Matching Kneser Graph Conjecture For High Chromatic Numbers

\noindent In this paper, we show that for any positive integers $r$, $k$, $\mathrm{\Theta }$, and $\mathrm{\Gamma }$ such that $k\ge 2$ and $r\ge k+\mathrm{\Gamma }$, there exists a connected graph $G$ for which $$\begin{array}{llcr}\omega (G)=\chi (G)=k,& \chi (G,r{K}_2)=\mathrm{\Theta },& \mathrm{a}\mathrm{n}\mathrm{d}& \mathrm{\mid }E(G)\mathrm{\mid }-\mathrm{e}\mathrm{x}(G,r{K}_2)=\mathrm{\Theta }+\mathrm{\Gamma }\mathrm{.}\end{array}$$