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

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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Discrete Mathematics, Elsevier, 2016, 339 (8)
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Soumis le : jeudi 28 avril 2016 - 19:09:11
Dernière modification le : samedi 30 avril 2016 - 01:06:36
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  • HAL Id : hal-01157902, version 2
  • ARXIV : 1505.07780

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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). <hal-01157902v2>

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