The Hadwiger number, chordal graphs and $ab$-perfection

A graph is chordal if every induced cycle has three vertices. The Hadwiger number is the order of the largest complete minor of a graph. We characterize the chordal graphs in terms of the Hadwiger number and we also characterize the families of graphs such that for each induced subgraph $H$, (1) the Hadwiger number of $H$ is equal to the maximum clique order of $H$, (2) the Hadwiger number of $H$ is equal to the achromatic number of $H$, (3) the $b$-chromatic number is equal to the pseudoachromatic number, (4) the pseudo-$b$-chromatic number is equal to the pseudoachromatic number, (5) the Hadwiger number of $H$ is equal to the Grundy number of $H$, and (6) the $b$-chromatic number is equal to the pseudo-Grundy number.