Hierarchical Colorings of Cographs

TL;DR

This paper explores hierarchical colorings of cographs, demonstrating that they generalize greedy colorings and also use no more than the chromatic number, with implications for graph coloring theory.

Contribution

It introduces hierarchical coloring as a generalization of greedy coloring for cographs, showing it also achieves the chromatic number efficiently.

Findings

Hierarchical coloring encompasses greedy coloring as a special case.

Hierarchical coloring requires no more than the chromatic number of the graph.

The approach has potential applications in reciprocal best match graphs.

Abstract

Cographs are exactly hereditarily well-colored graphs, i.e., the graphs for which a greedy coloring of every induced subgraph uses only the minimally necessary number of colors $\chi(G)$. In recent work on reciprocal best match graphs so-called hierarchically coloring play an important role. Here we show that greedy colorings are a special case of hierarchical coloring, which also require no more than $\chi(G)$ colors.