On the Δ-edge stability number of graphs

The $\Delta$-edge stability number ${\mathrm{es}}_{\Delta }\left(G\right)$ of a graph $G$ is the minimum number of edges of $G$ whose removal results in a subgraph $H$ with $\Delta \left(H\right)=\Delta \left(G\right)-1$. Sets whose removal results in a subgraph with smaller maximum degree are called mitigating sets. It is proved that there always exists a mitigating set which induces a disjoint union of paths of order 2 or 3. Minimum mitigating sets which induce matchings are characterized. It is proved that to obtain an upper bound of the form ${\mathrm{es}}_{\Delta }\left(G\right)\le c\left|V\left(G\right)\right|$ for an arbitrary graph $G$ of given maximum degree $\Delta$, where $c$ is a given constant, it suffices to prove the bound for $\Delta$-regular graphs. Sharp upper bounds of this form are derived for regular graphs. It is proved that if $\Delta \left(G\right)\ge \frac{\left|V\left(G\right)\right|-2}{3}$ or the induced subgraph on maximum degree vertices has a $\Delta \left(G\right)$-edge coloring, then ${\mathrm{es}}_{\Delta }\left(G\right)\le \lceil \left|V\left(G\right)\right|/2\rceil$.