The Grundy number of a graph G, denoted by Γ(G), is the largest k such that there exists a partition of V(G), into k independent sets <sup>V1</sup>,...,<sup>Vk</sup> and every vertex of <sup>Vi</sup> is adjacent to at least one vertex in <sup>Vj</sup>, for every j<i. The objects which are studied in this article are families of r-regular graphs such that Γ(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 <sup>C4</sup>, for r≤4. Furthermore, our propositions imply results on the partial Grundy number. © 2014 Elsevier B.V. All rights reserved.
Gastineau, N., Kheddouci, H., & Togni, O. (2014). On the family of r-regular graphs with Grundy number r + 1. Discrete Mathematics, 328(1), 5–15. https://doi.org/10.1016/j.disc.2014.03.023