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

1 GOAL - Graphes, AlgOrithmes et AppLications
LIRIS - Laboratoire d'InfoRmatique en Image et Systèmes d'information
Abstract : Gyárfás et al. and Zaker have proven that the Grundy number of a graph $G$ satisfies $\Gamma(G)\ge t$ 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 $\varphi(G)\ge t$ and $\partial\Gamma(G)\ge t$ (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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https://hal-univ-bourgogne.archives-ouvertes.fr/hal-01157902
Contributeur : Nicolas Gastineau <>
Soumis le : jeudi 28 avril 2016 - 19:09:11
Dernière modification le : lundi 16 décembre 2019 - 10:18:07
Archivage à long terme le : mardi 15 novembre 2016 - 17:38:30

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Brice Effantin, Nicolas Gastineau, Olivier Togni. A characterization of b-chromatic and partial Grundy numbers by induced subgraphs. Discrete Mathematics, Elsevier, 2016, 339 (8), pp.2157 - 2167. ⟨10.1016/j.disc.2016.03.011⟩. ⟨hal-01157902v2⟩

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