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

Nicolas Gastineau (1, 2) , Hamamache Kheddouci (1) , Olivier Togni (3)

1 GrAMA - Graphes, Algorithmes et Multi-Agents
2 GOAL - Graphes, AlgOrithmes et AppLications
3 Le2i - Laboratoire Electronique, Informatique et Image [UMR6306]

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.

Mots clés: False twins; Regular graph; Grundy number; Partial Grundy number

Domaines: Mathématique discrète [cs.DM]

Nicolas Gastineau, Hamamache Kheddouci, Olivier Togni. On the family of $r$-regular graphs with Grundy number $r+1$. Discrete Mathematics, 2014, 328 (5-15), pp.5-15. ⟨10.1016/j.disc.2014.03.023⟩. ⟨hal-00922022v2⟩