Let G be a non-empty graph. The chromatic edge-stability number of G is the minimum number of edges whose removal results in a spanning subgraph H with smaller chromatic number than G. The concept was introduced by Staton in 1980, but remained largely unnoticed for nearly four decades. In this talk, we present an overview of (mostly recent) results on the topic. In particular, we consider graphs that are critical for the chromatic edge-stability number. The variation of chromatic edge-stability concerning edge colorings is also presented, where the chromatic edge-stability index of G is the minimum number of edges whose removal results in a spanning subgraph H with smaller chromatic index than G.
