The Matching Kneser Graph Conjecture For High Chromatic Numbers

Abstract

\noindent In this paper, we show that for any positive integers $r$, $k$, $\Theta$, and $\Gamma$ such that $k \geq 2$ and $r \geq k + \Gamma$, there exists a connected graph $G$ for which $$\begin{array}{llcr} \omega (G) = \chi (G) = k, & \chi \left( G , rK_2 \right) = \Theta , & {\rm and} & |E(G)| - {\rm ex}\left( G , rK_2 \right) = \Theta + \Gamma . \end{array}$$