On quas-monotonous graphs

A dominating coloring by k colors is a proper k coloring where every color i has a representative vertex xi adjacent to at least one vertex in each of the other classes. The b-chromatic number, b(G), of a graph G is the largest integer k such that G admits a dominating coloring by k colors. A graph G = (V,E) is said b-monotonous if b(H1) ≥ b(H2) for every induced subgraph H1 of G and every subgraph H2 of H1. Here we say that a graph G is quasi b-monotonous , or simply quasi-monotonous, if for every vertex v ∈ V , b(G− v) ≤ b(G) + 1. We show study the quasi-monotonicity of several classes. We show in particular that chordal graphs are not quasi-monotonous in general,whereas chordal graphs with large b-chromatic number, and (P, coP, chair, cochair)-free graphs are quasimonotonous; (P5, coP5, P )-free graphs are monotonous.Finally we give new bounds for the b-chromatic number of any vertex deleted subgraph of a chordal graph.