Hierarchical and modularly-minimal vertex colorings

Abstract

Cographs are exactly the hereditarily well-colored graphs, i.e., the graphs for which a greedy vertex coloring of every induced subgraph uses only the minimally necessary number of colors χ(G). Greedy colorings are shown to be a subclass of hierarchical vertex colorings that naturally appear in phylogenetic combinatorics. The hierarchical colorings of cographs form a special class of minimal colorings. The fact that cotrees are the modular decomposition trees of cographs suggests a natural generalization: A coloring σ is modularly-minimal if |σ(M)| = χ(M) for every strong module M of G. We show that every graph admits a modularly minimal coloring. On cographs, furthermore, the modularly minimal colorings coincide with the hierarchical colorings. For certain hereditary graph classes efficient algorithms can be designed to compute modularly-minimal colorings.