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Article Dans Une Revue Discrete Mathematics Année : 2016

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

Brice Effantin
Nicolas Gastineau
Olivier Togni

#### Résumé

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 $7$.

#### Domaines

Informatique [cs] Mathématique discrète [cs.DM]

### Dates et versions

hal-01157902 , version 1 (28-05-2015)
hal-01157902 , version 2 (28-04-2016)

### Identifiants

• HAL Id : hal-01157902 , version 2
• ARXIV :
• DOI :

### Citer

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

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