On the Chromatic Edge Stability Number of Graphs

Abstract

The chromatic edge stability number $es _{\chi }(G)$ of a graph G is the minimum number of edges whose removal results in a graph $H \subseteq G$ with chromatic number $\chi (H) = \chi (G) - 1$. The chromatic bondage number $\rho (G)$ of G is the minimum number of edges between any two color classes in a $\chi (G)$-coloring of G, where the minimum is taken over all $\chi (G)$-colorings of G. In this paper, we characterize graphs for which these two parameters coincide. Moreover, we give general bounds and we determine these parameters for several classes of graphs.