The $\chi $-stability index $\mathrm {es}_{\chi }(G)$ of a graph G is the minimum number of its edges whose removal results in a graph with chromatic number smaller than that of G. We consider three open problems from Akbari et al. [‘Nordhaus–Gaddum and other bounds for the chromatic edge-stability number’, European J. Combin. 84 (2020), Article no. 103042]. We show by examples that a known characterisation of k-regular ($k\le 5$) graphs G with $\mathrm {es}_{\chi }(G) = 1$ does not extend to $k\ge 6$, and we characterise graphs G with $\chi (G)=3$ for which $\mathrm { es}_{\chi }(G)+\mathrm {es}_{\chi }(\overline {G}) = 2$. We derive necessary conditions on graphs G which attain a known upper bound on $\mathrm { es}_{\chi }(G)$ in terms of the order and the chromatic number of G and show that the conditions are sufficient when $n\equiv 2 \pmod 3$ and $\chi (G)=3$.