On minimally b-imperfect graphs

A b-coloring is a coloring of the vertices of a graph such that each color class contains a vertex that has a neighbour in all other color classes. The b-chromatic number of a graph GG is the largest integer kk such that GG admits a b-coloring with kk colors. A graph is b-perfect if the b-chromatic number is equal to the chromatic number for every induced subgraph HH of GG. A graph is minimally b-imperfect if it is not b-perfect and every proper induced subgraph is b-perfect. We give a list FF of minimally b-imperfect graphs, conjecture that a graph is b-perfect if and only if it does not contain a graph from this list as an induced subgraph, and prove this conjecture for diamond-free graphs, and graphs with chromatic number at most three.