# On the family of $r$-regular graphs with Grundy number $r+1$

* Auteur correspondant
1 GrAMA - Graphes, Algorithmes et Multi-Agents
LIRIS - Laboratoire d'InfoRmatique en Image et Systèmes d'information
Abstract : The Grundy number of a graph $G$, denoted by $\Gamma(G)$, is the largest $k$ such that there exists a partition of $V(G)$, into $k$ independent sets $V_1,\ldots, V_k$ and every vertex of $V_i$ is adjacent to at least one vertex in $V_j$, for every $j < i$. The objects which are studied in this article are families of $r$-regular graphs such that $\Gamma(G) = r + 1$. Using the notion of independent module, a characterization of this family is given for $r=3$. Moreover, we determine classes of graphs in this family, in particular the class of $r$-regular graphs without induced $C_4$, for $r \le 4$. Furthermore, our propositions imply results on partial Grundy number.
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Article dans une revue
Discrete Mathematics, Elsevier, 2014, 328 (5-15)

https://hal.archives-ouvertes.fr/hal-00922022
Contributeur : Nicolas Gastineau <>
Soumis le : lundi 19 mai 2014 - 17:46:38
Dernière modification le : vendredi 29 avril 2016 - 18:24:58
Document(s) archivé(s) le : lundi 10 avril 2017 - 23:52:04

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### Identifiants

• HAL Id : hal-00922022, version 2
• ARXIV : 1312.6503

### Citation

Nicolas Gastineau, Hamamache Kheddouci, Olivier Togni. On the family of $r$-regular graphs with Grundy number $r+1$. Discrete Mathematics, Elsevier, 2014, 328 (5-15). <hal-00922022v2>

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