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

## TL;DR

This paper characterizes chordal graphs and various graph families based on equalities involving the Hadwiger number and other graph invariants, providing new insights into graph structure and properties.

## Contribution

It introduces characterizations of chordal graphs and specific graph families through equalities involving the Hadwiger number and other invariants, expanding understanding of graph perfection.

## Key findings

- Chordal graphs characterized by Hadwiger number properties.
- Families of graphs where Hadwiger number equals clique or chromatic numbers.
- New relationships between Hadwiger number and various graph invariants.

## Abstract

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.