On the $\Delta$-edge stability number of graphs

Abstract

The $\Delta$-edge stability number ${\mathrm{es}}_{\Delta }(G)$ of a graph $G$ is the minimum number of edges of $G$ whose removal results in a subgraph $H$ with $\Delta (H)=\Delta (G)-1$. 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 ${\mathrm{es}}_{\Delta }(G)\le c|V(G)|$ 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 $\Delta (G)\ge \frac{|V(G)|-2}{3}$ or the induced subgraph on maximum degree vertices has a $\Delta (G)$-edge coloring, then ${\mathrm{es}}_{\Delta }(G)\le \lceil |V(G)|/2\rceil$.