# Critical graphs for the chromatic edge-stability number

## TL;DR

This paper studies the properties of graphs that are critical with respect to their chromatic edge-stability number, classifies certain types, and relates the problem to graph coloring decision problems.

## Contribution

It introduces and classifies $(3,2)$-critical graphs with limited odd cycles and links the criticality decision problem to polynomial-time graph coloring.

## Key findings

- Classified $(3,2)$-critical graphs with up to four odd cycles.
- Proved polynomial-time reduction from chromatic criticality to $(k,2)$-criticality decision.
- Established properties of edge-stability critical graphs.

## Abstract

The chromatic edge-stability number ${\rm es}_{\chi}(G)$ of a graph $G$ is the minimum number of edges whose removal results in a spanning subgraph $G'$ with $\chi(G')=\chi(G)-1$. Edge-stability critical graphs are introduced as the graphs $G$ with the property that ${\rm es}_{\chi}(G-e) < {\rm es}_{\chi}(G)$ holds for every edge $e\in E(G)$. If $G$ is an edge-stability critical graph with $\chi(G)=k$ and ${\rm es}_{\chi}(G)=\ell$, then $G$ is $(k,\ell)$-critical. Graphs which are $(3,2)$-critical and contain at most four odd cycles are classified. It is also proved that the problem of deciding whether a graph $G$ has $\chi(G)=k$ and is critical for the chromatic number can be reduced in polynomial time to the problem of deciding whether a graph is $(k,2)$-critical.