  The $\chi$-stability index ${\rm es}_{\chi}(G)$ of a graph $G$ is the minimum
number of its edges whose removal results in a graph with the chromatic number
smaller than that of $G$. In this paper three open problems from [European J.\
Combin.\ 84 (2020) 103042] are considered. Examples are constructed which
demonstrate that a known characterization of $k$-regular ($k\le 5$) graphs $G$
with ${\rm es}_{\chi}(G) = 1$ does not extend to $k\ge 6$. Graphs $G$ with
$\chi(G)=3$ for which ${\rm es}_{\chi}(G)+{\rm es}_{\chi}(\overline{G}) = 2$
holds are characterized. Necessary conditions on graphs $G$ which attain a
known upper bound on ${\rm es}_{\chi}(G)$ in terms of the order and the
chromatic number of $G$ are derived. The conditions are proved to be sufficient
when $n\equiv 2 \pmod 3$ and $\chi(G)=3$.
