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.