For any set $X$, let $[X]^2= \big\{\{x,y\}: x\neq y \in X\big\}$. Let $\kappa$ be an infinite cardinal. Let $G_\kappa = ([\kappa]^2, E_\kappa)$ where $E_\kappa = \big\{\{a,b\}\in \big[[\kappa]^2\big]^2: \max(a) = \min(b)\big\}$.

In $\newcommand{\ZFC}{{\sf (ZFC)}}\ZFC$, is $\chi(G_\kappa) = \kappa$ for all infinite cardinals $\kappa$?