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

Brice Effantin 1, * Nicolas Gastineau 2, * Olivier Togni 2, *
* Auteur correspondant
1 GOAL - Graphes, AlgOrithmes et AppLications
LIRIS - Laboratoire d'InfoRmatique en Image et Systèmes d'information
Abstract : Gyarfas et al. and Zaker have proven that the Grundy number of a graph G satisfies Gamma(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 phi(G) >= t and partial derivative Gamma(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. (C) 2016 Elsevier B.V. All rights reserved.
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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. 〈http://www.sciencedirect.com/science/article/pii/S0012365X16300565〉. 〈10.1016/j.disc.2016.03.011〉. 〈hal-01434807〉



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