On the $\Delta$-edge stability number of graphsAkbari, Saieed (Avtor) Hosseini Dolatabadi, Reza (Avtor) Jamaali, Mohsen (Avtor) Klavžar, Sandi (Avtor) Movarraei, Nazanin (Avtor) vertex degree▫$\Delta$▫-edge stability numbermatchingedge coloringThe $\Delta$-edge stability number ${\rm es}_{\Delta}(G)$ 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 ${\rm es}_{\Delta}(G) \leq c |V(G)|$ 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 $\Delta(G) \geq\frac{|V(G)|-2}{3}$ or the induced subgraph on maximum degree vertices has a $\Delta(G)$-edge coloring, then ${\rm es}_{\Delta}(G) \le {\lceil |V(G)|/2\rceil}$.20252025-05-07 10:38:17Neznano22176sl