Characterization of some b-chromatic edge critical graphs

A b-coloring is a proper coloring of the vertices of a graph such that each color class has a vertex that is adjacent to a vertex of every other color. The b-chromatic number b(G) of a graph G is the largest k such that G admits a b-coloring with k colors. A graph G is b-chromatic edge critical if for any edge e of G, the b-chromatic number of G − e is less than the b-chromatic number of G. We call these graphs b-chromatic edge critical or just edge b-critical. We show that any edge b-critical graph G satisfies b(G) = Δ(G) + 1, where Δ(G) is the maximum degree of G, and we characterize edge b-critical P4 sparse graphs and edge b-critical quasi-line graphs.