Nordhaus–Gaddum and other bounds for the chromatic edge-stability number

Abstract

Let G be a graph. The chromatic edge-stability number esχ(G) of a graph G is the minimum number of edges of G whose removal results in a graph H with χ(H)=χ(G)−1. A Nordhaus–Gaddum type inequality for the chromatic edge-stability number is proved. Sharp upper bounds on esχ are given for general graphs in terms of size and of maximum degree, respectively. All bounds are demonstrated to be sharp. Graphs with esχ=1 are considered and in particular characterized among k-regular graphs for k≤5. Several open problems are also stated.