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Pré-Publication, Document De Travail Année : 2015

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

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 $7$.
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Dates et versions

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

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Brice Effantin, Nicolas Gastineau, Olivier Togni. A characterization of b-chromatic and partial Grundy numbers by induced subgraphs. 2015. ⟨hal-01157902v1⟩
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