  The chromatic edge stability index $\mathrm{es}_{\chi'}(G)$ of a graph $G$ is
the minimum number of edges whose removal results in a graph with smaller
chromatic index. We give best-possible upper bounds on $\mathrm{es}_{\chi'}(G)$
in terms of the number of vertices of degree $\Delta(G)$ (if $G$ is Class 2),
and the numbers of vertices of degree $\Delta(G)$ and ${\Delta(G)-1}$ (if $G$
is Class 1). If $G$ is bipartite we give an exact expression for
$\mathrm{es}_{\chi'}(G)$ involving the maximum size of a matching in the
subgraph induced by vertices of degree $\Delta(G)$. Finally, we consider
whether a minimum mitigating set, that is a set of size
$\mathrm{es}_{\chi'}(G)$ whose removal reduces the chromatic index, has the
property that every edge meets a vertex of degree at least $\Delta(G)-1$; we
prove that this is true for some minimum mitigating set of $G$, but not
necessarily for every minimum mitigating set of $G$.

Combinatorics