A note on the orientation covering number

Given a graph $G$, its orientation covering number $\sigma(G)$ is the smallest non-negative integer $k$ with property that we can choose orientations of $G$ such whenever $x, y, z$ are vertices $xy,xz\in E(G)$ then there chosen in which both $xy$ and $xz$ oriented away from $x$. Esperet, Gimbel King showed $\sigma(G)\leq \sigma\left(K_{\chi(G)}\right)$, where $\chi(G)$ chromatic asked whether always have equality. In this note prove it indeed case $\sigma(G)=\sigma(K_{\chi(G)})$. We also determine exact value $\sigma(K_n)$ explicitly for `most' values $n$.