A characterization of b-chromatic and partial Grundy numbers by induced subgraphs

Gyarfas et?al. and Zaker have proven that the Grundy number of a graph G satisfies ? ( G ) ? t if and only if G contains an induced subgraph called a t -atom. The family of t -atoms has bounded order and contains a finite number of graphs. In this article, we introduce equivalents of t -atoms for b-coloring and partial Grundy coloring. This concept is used to prove that determining if ? ( G ) ? t and ? ? ( G ) ? t (under conditions for the b-coloring), for a graph G , is in XP with parameter t . We illustrate the utility of the concept of t -atoms by giving results on b-critical vertices and edges, on b-perfect graphs and on graphs of girth at least 7.

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